Table of contents (arranged alphabetically)

Adding with decimals (9) Benchmark numbers (14) Circles (22) Combining like terms (17) Common reasons for setting up a proportion (4) Common reasons for using the operations (3) Converting between fractions, decimals, and percents (15) Distribution (17) Divisibility (5) Factoring (17) Fractions (14) Greatest common factor (7) Inequalities (20) Invisible things (14) Least common multiple (8) Long division (13) Measures of center, spread, and mode (23) Multiplication and division tables (5) Multiplying and dividing by powers of ten (4) Multiplying with decimals – matrix method (11) Multiplying with decimals – traditional method (12) Negatives (17) Percent expressions (16) Percent word problems (15) Place value (5) Probability (21) Properties (3) Proportional relationships (16) Representing division (4) Rounding (6) Scale drawings (18) Similar figures (20) Solving complicated equations (19) Solving simple equations (18) Subtracting with decimals (10) Surveys (24) The 5 actions of arithmetic (2) Unit rates (16) Writing equations and inequalities (25)

Reference Pages

The 5 Actions of Arithmetic (yes, five!)

The commutative property still holds for fractions.

Challenge question: Can you show that 12

× 34=3

4× 1

2 ?

The commutative property DOES NOT hold.

The commutative property DOES NOT hold.

“Fitting into” always gives you the same numerical answer as “splitting up” so people just use the ÷ symbol for everything. That’s a little confusing because whenever you see the ÷ symbol, it could mean splitting up but it could also mean fitting into. Notice that when we’re splitting up, the answer refers to the number of items in each group but when we’re fitting into, the answer refers to the number of groups.

Put together +¿

Take away −¿

Group ×

Split up ÷

Fit into

5+3=83+5=8Five put together with three Three put together with five

The commutative property holds.

5−3=2 3−5=−2 Three taken away from five Five taken away from three

The commutative property DOES NOT hold.

5×3=15 3× 5=15 Five groups of three Three groups of five

3+3+3+3+3=15 5+5+5=15

The commutative property holds.

When the number of groups is a whole number, grouping can be thought of as repeated addition like in the examples above and the first example below…

5 × 34=15

4 Five groups of three fourths.

When the number of groups is a fraction , grouping can be thought of as taking a piece of a number like in the example below…

34

×5=154

Three fourths of a group of five.

+ + + +

Start with five. Split it up into four parts and take three of them.

8 ÷ 2=4 items per group 2 ÷ 8=14 of an item per group

Eight split up Two split up into two groups into eight groups

1 2 3 4 14

1 2 3 4

8 2=4 groups 28= 14 of a group

How many groups How many groups of two fit into eight? of eight fit into two?

14

a

Commutative property: a+b=b+a a× b=b× a

Switching the order for addition or multiplication does not change the result.Switching the order for subtraction gives you the opposite (commutative property does not hold).Switching the order for division gives you the reciprocal (commutative property does not hold).

Examples: 12+4=16 12 ×4=48 12−4=8 12 ÷ 4=3

4+12=16 4 ×12=48 4−12=−8 4 ÷12=13

Associative property: (a+b )+c=a+(b+c ) (a × b ) ×c=a × (b × c )

When adding or multiplying a string of numbers, we may group the numbers in any way without changing the result. The associative property does not hold for subtraction or division.

Examples: (10+5 )+4=15+4=19 (10 ×5 ) × 4=50× 4=200 10+(5+4 )=10+9=19 10 ×(5× 4 )=10 ×20=200

(10−5 )−4=5−4=1 (10 ÷ 5 )÷ 4=2÷ 4=12

10−(5−4 )=10−1=9 10 ÷(5 ÷ 4)=10÷ 1.25=8

Distributive property: a × (b+c )=a ×b+a× c

The products of sums is equal to the sums of products.

Common reasons for subtraction

Taking away (by definition)Finding how many more of something there is compared to something else.Finding how much greater or longer something is compared to something else.Finding the distance between two points.

Common reasons for divisionSplitting up into equal parts (by definition)Seeing how many times one thing fits into another (by definition)Finding a unit rate

Common reasons for multiplicationGrouping as in repeated addition (by definition)Finding a fraction or percent of a number (by definition)Finding the area of a parallelogramFinding the volume of a rectangular prism

Common reasons for additionPutting together (by definition)Finding a total

Common reasons for using the operations

Properties

Example: 3× (6+2 )=3×8=24 3×6+3× 2=18+6=24

Converting a fraction into a percent

Numerator

Denominator= Percent (x )

100

Find the partFind the wholeFind the percent of the whole

Part

Whole=Percent

100

Find the percent of change/error

difference

original /real=

Percent (x )100

Find the new valueFind the original value

new

original=100 ± percent

100

Unit rateExample: When looking for miles per hour, miles go in the numerator and hours go in the denominator.

MilesHours

=Unit rate(x)

1

Basic proportionConverting units

Small Unit ASmallUnit B

=BigUnit ABigUnit B

Small Unit ABig Unit A

=Small Unit BBigUnit B

Similar figuresHere “small” and “big” refer to the small shape and big shape.

Small Side ASmall Side B

=Big Side ABig Side B

Small Side ABig Side A

=Small Side BBig B

Probability: Making predictions

GoodTotal

=Predition(x )Attempts

Surveys

Good

Total(sample)=

Prediction(x )Total (population)

Common reasons for setting up a proportion

Multiplying and dividing by powers of 10

Representing division

There are three ways of writing division. To convert between them, Simply FoLlow the rainbow. Make sure your rainbow has the Standard division symbol on the left, Fraction bar in the middle, and Long division symbol on the right. Make sure you also know which number is the dividend, which number is the divisor, and all the different ways of writing the division in words. Be careful! With certain phrases you say the dividend first and with others you say the divisor first.

82

8 ÷ 2 2 ) 8Standard Fraction Long

28

2 ÷ 8 8 ) 2

Dividend: 8 Divisor: 2What is 8 divided by 2? What is 2 divided into 8?What is 8 split up into 2 equal parts? How many times can 2 fit into 8?

Dividend: 2 Divisor: 8What is 2 divided by 8? What is 8 divided into 2?What is 2 split up into 8 equal parts? How many times can 8 fit into 2?

Answer:

14

Answer:4

To solve proportions…

25= x

6

12=5 x

12=5 x 5 5

2.4 = x

Cross multiply.

Put an equal sign between the products.

Divide both sides by whatever is next to the unknown.

There are three ways of writing division. To convert between them, Simply FoLlow the rainbow. Make sure your rainbow has the Standard division symbol on the left, Fraction bar in the middle, and Long division symbol on the right. Make sure you also know which number is the dividend, which number is the divisor, and all the different ways of writing the division in words. Be careful! With certain phrases you say the dividend first and with others you say the divisor first.

Multiplication and division tables

Divisibility

To tell if a number is divisible by…

2See if it ends in 0, 2, 4, 6, or 8.

3Add up all the digits and see if you get a number that is divisible by 3. For example, to check 825, do 8+2+5=15. 15 is divisible by 3 so the original number 825 is as well.

4Check the two-digit number at the end and see if it is divisible by 4. For example, to check 97,005,328, just check 28. 28 is divisible by 4 so the original number 97,005,328 is as well.

5See if it ends in 0 or 5.

6Make sure it is divisible by 2 and 3. For example, 34 is divisible by 2 because it ends in 4. To check if it’s divisible by 3, do 3+4=7. 7 is not divisible by 3 so 34 is not divisible by 3 or 6.

8Check the three-digit number at the end and see if it is divisible by 8. For example, to check 4,260,520, just check 520. 520 is divisible by 8 (you can check that with long division) so the original number 4,260,520 is as well.

10See if it ends in 0.

9Add up all the digits and see if you get a number that is divisible by 9. For example, to check 559, do 5+5+9=19. 19 is not divisible by 9 so the original number 559 is not either.

Place value

4 0 3 , 9 7 4 . 0 5 5 6 0

100,000’s 10,000’s 1,000’s 100’s 10’s 1’s 1

10’s

1100

’s 1

1,000’s

110,000

’s

1100,000 ’s

Place values are based on how many spaces away from the decimal point a digit is.

If we don’t care about being exactly right, then we can round a number to get another number which is close enough.

Rounding

Step 1:

R

587,403.92

R

16.04281

Example 1:

Round 587,403.92 to the nearest thousand.

Example 2:

Round 16.04281 to the nearest thousandth.

Step 2:

RD

587,403.92

RD

16.04281

Step 3: If the decision digit is 0, 1, 2, 3, or 4 then the rounding digit stays the same (Notice the 7 is still a 7).

RD

587,403.92

It the decision digit is 5, 6, 7, 8, or 9 then the rounding digit gets bumped up by 1. (Notice the 2 became a 3).

RD

16.04381

16.043

Step 4:

587,000

Find the digit in the place value you’re rounding to. Call it the “rounding digit”.

Find the digit one space to the right of the rounding digit. Call it the “decision digit”.

Remove all digits after the rounding digit (even the decision digit). More specifically, digits after the rounding digit but before the decimal point become zeros. Digits after the rounding digit and after the decimal point get removed completely.

If you have to bump up a 9, it becomes a 0 and the next digit on the left gets bumped up by one.

Example 3:Round 38,497.20581 to the nearest tens place.

RD The result is…

38,497.20581 38,500

Rounding to the nearest “whole number”, “percent”, or “dollar” means rounding to the ones place.Rounding to the nearest “cent” means rounding to the hundredths place because a penny is a hundredth of a dollar.

If you have to bump up a 9 and the digits directly to the left of it are also 9’s then they all become zeros and the digit to their left gets bumped up by one.

Example 4:Round 999,959.99546 to the nearest hundredths place.

RD The result is…

999,959.99546 999,960.00

Trouble with 9’s

Round to what?

162

2 81

9 9

3 3 3 3

126

2 63

7 9

3 3

The greatest common factor of two numbers is the biggest number that goes nicely into both of them. Note: A “factor” of a number is a number that goes into it.

Simple method: This method is better when the numbers are small. Sometimes you can even do it in your head.

Example: Find the greatest common factor of 24 and 36. Step 1: List the factors of each number. 24: 1 2 3 4 6 8 12 24 36: 1 2 3 4 6 9 12 18 36

Step 2: Circle the factors they have in common. 24: 1 2 3 4 6 8 12 24 36: 1 2 3 4 6 9 12 18 36

Step 3: Choose the greatest (biggest) common factor from Step 2. It’s 12!

Prime factorization method: This method is better when the numbers are larger. The idea here is to build the biggest factor we can by using as many factors as possible. However, we can’t use all the factors because not all of them are factors of every number.

Example: Find the greatest common factor of 126 and 162.

Step 1: Find the prime factorization of each number. (Just keep breaking down the numbers into factors until you can’t break them down anymore.)

Step 2: Find the factors that both numbers have in common. Pay attention to how many of them there are.

Both numbers have at least ONE 2. Both numbers have at least TWO 3’s.

Step 3: Multiply the factors from the last step together. 2 ×3 ×3=18.

Shortcut method: If one of the numbers is a factor of the other one then that number is the greatest common factor.

Example: Find the greatest common factor of 10 and 20.

10 is a factor of 20 so the GCF is 10.

Greatest common factor (GCF)

The least common multiple of two numbers is the smallest number that is a multiple of both of them. Note: A “multiple” of a number is a number that it goes into.

Simple method: This method is better when the numbers are small. Sometimes you can even do it in your head.

Example: Find the least common multiple of 2 and 5. Step 1: List the multiples of each number. 2: 2 4 6 8 10 12 14 16 18 20 22… 5: 5 10 15 20 25 30…

Step 2: Circle the multiples they have in common. 2: 2 4 6 8 10 12 14 16 18 20 22… 5: 5 10 15 20 25…

Step 3: Choose the least (smallest) common multiple from Step 2. It’s 10!

Prime factorization method: The idea here is to use as few factors as possible to still end up with a multiple of all the numbers. If you do it correctly then each number’s prime factorization will be found within the prime factorization of the LCM.

Example: Find the greatest common factor of 150 and 48.

Step 1: Find the prime factorization of each number. (Just keep breaking down the numbers into factors until you can’t break them down anymore.)

Step 2: For each factor that you see, write the largest number of times it occurs in any one number. The largest number of 2’s is FOUR (48 has this). The largest number of 5’s is TWO (150 has this). The largest number of 3’s is ONE (both have this).

Step 3: Multiply the factors from the last step together. 2 ×2×2×2 ×5×5 ×3=1,200.

*If there are no common factors in the prime factorizations then you can skip steps 2 and 3 and just multiply the original numbers together.

Shortcut method: If one of the numbers is a multiple of the other one then that number is the least common multiple.Example: Find the least common multiple of 10 and 20.

Least common multiple (LCM)

150

3 50

2 25

5 5

48

2 24

2 12

3 4

2 2

20 is a multiple of 10 so the LCM is 20.When adding with decimals, keep in mind…

1. The largest number does NOT have to go on top like it does when subtracting.

2. The decimals MUST be lined up, including in the answer. If you don’t see a decimal, it’s invisible. It’s at the very end of the number.

3. Blank spaces count as zeroes. You may put them in if you like, but it is not as important as it is with subtraction.

4. Always add from right to left.

5. If you get a two-digit sum, add the first digit to the next column and write the second digit below. Once you get to the end, just write everything below.

Example:

59 . 9 988 .+ 6 . 74 .

Line up decimals.59.9 + 988 + 6.74

Invisible decimal

59 . 9 1+9+8+6=24 988 .+ 6 . 74

59 . 9 9+7=16 988 .+ 6 . 74

59 . 9 4+0=4 988 .+ 6 . 74

59 . 9 1+9=10 988 .+ 6 . 74

59 . 9 2+5+8=15 988 .+ 6 . 74

Adding with Decimals

Subtracting with Decimals

Example 1:

Example 2:

When subtracting, keep in mind…

1. The larger number MUST go on top. The larger number is not necessarily the number with the most digits. For example…

2. The decimals MUST be lined up, including in the answer. If you don’t see a decimal, it’s invisible. It’s at the very end of the number.

3. Blank spaces count as zeroes so put them in.

4. Always subtract from right to left.

5. We cannot take a larger number away from a smaller number. If this happens, we must “borrow” from the next column by reducing the digit in that column by one and then placing a 1 in front of the number that was too small in the first place (by doing this we are really adding 10 to the number that was too small).

6. We cannot borrow from 0. If this happens, we must instead borrow from the number in the next column, then put a 1 in front of the 0, turning it into a 10, then borrow from the 10 that we just created, turning it into a 9.

05

02

06

20

08

24

35

14

42

05

02

06

20

08

24

35

14

42

05

02

06

20

08

24

35

14

42

6 . 2 5

06

240

9

104

01

5

06

240

9

104

01

5

06

240

9

104

01

5

3 8 2

6 . 2 5

6 . 2 5

6 . 2 5 3 8 2

3 8 2

3 8 2

7

4

1

.

6 . 2 5

Step 1: Write one number horizontally and the other vertically (it doesn’t matter which number goes where) and draw boxes inside the two numbers. Each digit gets its own column or row of boxes. Each decimal should line up with a line.

Step 2: Draw diagonal lines going through the boxes. Each box should be split up into an upper left half and a lower right half.

Step 3: Multiply each digit of the horizontal number with each digit of the vertical number. Write the product inside the box. If the product only has one digit, put a 0 inside the upper left half of the box.

Step 4: Add the digits along each diagonal. This is just like normal addition (from right to left) so, for example, if you got a sum of 17, you would have to carry the 1 into the next diagonal.

Step 5: Draw a line down from the decimal in the horizontal number and to the left from the decimal in the vertical number. When the lines connect, continue along that diagonal. That’s where you will place the decimal in your final product. IMPORTANT: If you don’t see a decimal, it’s invisible. It comes at the end of the number and it’s a good idea to write it in. See Example 2.

5

3

2

02 4 6

1 1

2

02 4 6

1 1

5

3

5

3

3 8 2

7

4

1

.

5

4

6

31 2

21

5

4

6

31 2

21

7

4

1

.

7

4

5

3

1

.

7

4

5

3

1

.

Example 1: 6.25 ×7.41

Example 2: 382 ×5.3

Multiplying with Decimals – Matrix Method

.

.

.

.

.

.

Invisible decimalwritten in

Answer: 2024.6Answer: 46.3125

When multiplying with decimals, keep in mind…

1. You do NOT have to line up the decimals. In fact, it’s easier if you don’t.

2. Multiply from right to left. Multiply each digit in the bottom with each digit in the top. If you get a two-digit number as your product, put the first digit above the next number in the top and write the second digit on the bottom. If you put a digit in the top, add it to the next product. When you come to the end, just write everything in the bottom.

3. Whenever you move to the next digit in the bottom number, put an extra x (first 0, then 1, then 2, then 3, etc.) down below.

5. Once you’re done with the multiplying part, add everything up. Remember that x’s count as 0’s. After that, count how many spaces in from the right side the decimal was in the original numbers. Add the number of spaces together. That’s how many spaces from the right the decimal should be in the final answer.

6. It is easier to multiply by putting the number with more digits on top and the number with less digits below. You get the same answer no matter what, but you can see from the examples below, you’ll do a lot more work if you put the number with more digits on the bottom.

Multiplying with Decimals – Traditional Method

Repeat the four basic steps of long division until there are no more digits to bring down AND you got a 0 the last time you subtracted.

1. into 2. × 3. −¿ 4.

If there are no more digits to bring down but you didn’t get a 0 the last time you subtracted, place a decimal in the dividend and quotient and keep bringing down 0s until you’re done.

Didn’t get a 0

Bring down a 0.

Bring down another 0.

Now we’re done.

The decimal goes in the same place in the dividend and quotient.

Some students are afraid of 0s. Don’t be! If you get a 0 in Step 1, just put it in there like any other number.

Step 1: How many times can 9 fit into 4? 0 times. Write that next to the 1.

Make sure you’re putting your digits in the correct place in the quotient. In the example below, 25 does not go into 1 or 13, but it goes into 137 five times. The 5 would go above the 7, not above the 1 or 3.

A decimal in the dividend or quotient is alright, but we don’t want decimals in the divisor. If you see one, the first thing you should do is move it to the end. Next, however many spaces you moved it in the divisor, you have to move it that many spaces in the dividend as well.

If we see that one or more digits in the quotient are going to keep repeating themselves forever, we write a line over the repeating part in our answer.

Answer: 8.83 Answer: 0.75

Long Division

Invisible thingsIT1. Invisible decimalsIf you don’t see a decimal, it’s at the end of the number.

47 47.

IT2. Invisible coefficientsIf you don’t see a coefficient, it’s 1.

x1 x

IT3. Invisible exponentsIf you don’t see an exponent, it’s 1.

5 r 5 r1

IT4. Invisible operationsIt is often useful to think of a positive number as having a plus sign in front of it.

7 j−10 y+7 j−10 y

Benchmark numbersBE1. Half

12=0.5

BE2. Thirds

13=0.3

23=0.6

BE3. FourthsThink of quarters (25 cents each).

14=0.25

24=0.5

34=0.75

BE4. Fifths

15=0.2

25=0.4

35=0.6 4

5=0.8

BE5. SixthsYou can remember some of these more easily by reducing them.

16=0.16

26=1

3=0.3

36=1

2=0.5

46=2

3=0.6

56=0.83

BE6. EighthsYou can remember some of these more easily by reducing them.

18=0.125

28=1

4=0.25

38=0.375

48=1

2=0.5

58=0.625

68=3

4=0.75

BE7. NinthsThe ninths are easy. It’s just whatever the numerator is but repeating.

19=0.1

29=0.2

BE8. TenthsThe tenths are easy. It’s just whatever the numerator is.

1

10=0.1

210

=0.2

3

10=0.3 and so on…

Fractions

FR1. Add or subtractMake sure the denominators are the same first.

56+ 1

4 = 10

12+ 3

12=13

12

56+ 1

4 = 1012

− 312

= 712

FR2. MultiplyMultiply numerators and multiply denominators.

FR3. DivideMultiply by the reciprocal (keep- change-flip).

FR4. Mixed number to improper fractionMultiply denominator by whole number, add the numerator, keep the original denominator.

FR5. Improper fraction to mixed numberDivide the numerator by the denominator. The remainder is the new numerator. Keep the original denominator.

x2 x3 x2 x3

x3 x3

x2 x2

FR6. BorrowingSimilar to borrowing with decimals but you have to add the denominator instead of putting a 1 next to the number.

FR7. Multiplying or dividing with mixed numbersTurn them into improper fractions first.

FR8. Multiplying a mixed number with a whole numberYou could turn the mixed number into an improper fraction but sometimes it’s easier to distribute.

3×6 417

=18 1217

What are equivalent fractions? Equivalent fractions are fractions that have the same value, even though they look different.

Creating equivalent fractions / reducingIf you multiply the numerator and denominator of a fraction by the same number, the result is equivalent to the fraction you started with. This is because you really multiplied by 1!

27

× 55=10

35

The same is true when you divide and then it is called “reducing”.

2842

÷ 1414

=23

This is just 1!

This is just 1!

How do you know if two fractions are equivalent?Strategy 1: Divide the numerator by the denominator for each fraction and see if you get the same number.Strategy 2: Put the fractions next to each other and see if you can multiply or divide horizontally or vertically by the same number.Examples for Strategy 2…

What missing number makes two fractions equivalent?Strategy 1: Cross multiply and then divide both sides to isolate the unknown.Strategy 2: If you have numbers that go nicely into each other, you can symmetrically multiply or divide either horizontally or vertically.Examples for Strategy 2…

165

CO4. Decimal to peRcentMove the decimal point two spaces to the right.

CO2. Percent to decimaLMove the decimal point two spaces to the left.

PW4a. Find the new value (basic strategy)Step 1: Multiply the original value by the percent of change (written as a decimal).Step 2: Add or subtract the result (depending on the problem) from the original value.

Step 1: Step 2:

PW4b. Find the new value (alternate strategy)Step 1: Add or subtract the percent of change (depending on the problem) from 100%.Step 2: Multiply the result (written as a decimal) by the original value.

Step 1: Step 2:

PW2. Find the wholeDivide the part by the percent (written as a decimal).

100% ? mL

48 mL60%

0% 0 mL

PW1. Find the partMultiply the whole by the percent (written as a decimal).

100% 80 mL? mL60%

0% 0 mL

PW6a. Find the percent of changeStep 1: Subtract the two values.Step 2: Divide the difference by the original value (and convert into a percent).

Step 1: Step 2:

PW6b. Find the percent errorThese are very similar to “find the percent of change” problems except you would be given an estimate and a real value.

Step 1: Subtract the two values.Step 2: Divide the difference by the real value(and convert into a percent).

PW3. Find the percent of the wholeDivide the part by the whole and the convert into a percent.

0% 0 mL

?% 48 mL

80 mL100%

PW5a. Find the original valueStep 1: Add or subtract the percent of change (depending on the problem) from 100%.Step 2: Divide the result (written as a decimal) by the new value.

Step 1: Step 2:

PW5b. Using the P4a strategy backwards doesn’t work!A lot of students try to answer “find the original value” problems by using the P4a strategy in reverse. It won’t work.

Step 1: Step 2: wrong answer

BE5. SixthsYou can remember some of these more easily by reducing them.

16=0.16

26=1

3=0.3

36=1

2=0.5

46=2

3=0.6

56=0.83

BE6. EighthsYou can remember some of these more easily by reducing them.

18=0.125

28=1

4=0.25

38=0.375

48=1

2=0.5

58=0.625

68=3

4=0.75

BE7. NinthsThe ninths are easy. It’s just whatever the numerator is but repeating.

19=0.1

29=0.2

BE8. TenthsThe tenths are easy. It’s just whatever the numerator is.

1

10=0.1

210

=0.2

3

10=0.3 and so on…

Converting between fractions, decimals, and percents

CO1. Percent to fractionPut it over 100. Keep the decimal point where it was.

CO5. Fraction to decimalDivide the numerator by the denominator.

CO3a. Decimal to fractionPut it over the power of ten represented by the right-most place value. Do not keep the decimal point.

85.3=85310

85.73=8,573100

CO3b. Decimal to mixed numberSimilar to “decimal to fraction” but the number in front of the decimal will also go in front of the fraction.

85.3=85 310

85.73=85 73100

Percent word problems

Percent expressions

0% 60% 100% 140% 200%

PR4. How to tell if a graph is proportional

The line must be straight AND it must go through the origin (0,0).

UR5a. How to find a unit rate from a proportional graph: Dividing coordinates strategyPick any point on the line (except the origin) and divide the y-coordinate by the x-coordinate.

UR3. To calculate a unit rate

Divide!

But be careful of the order. To calculate miles per hour, divide miles by hours. To calculate hours per mile, divide hours by miles.

UR5b. How to find a unit rate from a proportional graph: (1,r) strategyFind 1 on the x-axis and draw a line straight up until you hit the line. The y-coordinate of that point is the unit rate.

UR6. How to find a unit rate from an equation

The number that is next to the variable is the unit rate

Equation Unit rate

y=7 x 7

y=−7 x −7

y=35

x 35

y=−35

x −35

Proportional relationships / Unit ratesPR1. Definition of proportionalityTwo quantities are in a proportional relationship if the ratio between them is always the same.

PR2. Consequence of proportionalityIf two quantities are in a proportional relationship, then if one gets multiplied by a number the other will get multiplied by the same number.

PR3. How to tell if a table is proportionalReduce the numbers in each row. If you always get the same number then the relationship is proportional.

You can also divide the numbers in each row. If you always get the same number, the relationship is proportional.

PR5. How to tell if an equation is proportionalTo represent a proportional relationship, an equation should not have more than one term on a side. Also, if one of the variables is in the denominator of a fraction, the equation is not proportional. NotProportional Proportional

y=−4 x y=−4 x+6

y= x10 y=10

x

UR1. Definition of a unit rate

A unit rate is a ratio between two quantities with one of the quantities having a value of 1. For example, 25 miles per hour means I drive 25 miles for every 1 hour that goes by.

UR2. Constant of proportionality

For proportional relationships, “constant of proportionality” means the same thing as unit rate.

UR4. How to find a unit rate from a proportional table

Pick any row and divide the numbers. Order matters!

10 dogs÷ 6 cats=1 23

dogs per cat

Of / As Much As(follows one-term form)140% of a number140% as much as a number60% of a number60% as much as a number

NE4. Finding the distance between two points on a number lineUse subtraction and absolute value.

Distance between dog and shirt:|5−−2|=|5+2|=|7|=7 or |−2−5|=|−7|=7 Distance between house and shirt:|−5−−2|=|−5+2|=|−3|=3 or |−2−−5|=|−2+5|=|3|=3

NE5. Signs of fractions

NE3. Multiplication / Division rules

Same signs (both positive or both negative): The answer will be positive.

Different signs (one positive and one negative): The answer will be negative.

NE1. Rewriting rules NE2. Adding / Subtracting a positive rules.

*Use these rules after rewriting.

**These rules are most useful when the first number is negative.

Same symbols: Add and keep the symbol.

Different symbols: Subtract and take the symbol of the “larger” number (the number with the greater absolute value).

* Note When doing subtraction, always put the larger number on top, even if it was negative in the expression.

Negatives

CT1. What are like terms?

Like terms are terms that all have the exact same variable(s) raised to the same exponent(s) or terms that all do not have a variable.

DI1. Distribution Multiply the number next to the parentheses with every term inside the parentheses. Pay very close attention to signs. Treat plus signs as positives and minus signs as negatives.Example 1: Example 2:

DI2. Distributing just a plus sign or minus signWhen a plus sign is next to the parentheses, nothing happens to the terms inside. When a minus sign is next to the parentheses, all of the terms inside change sign. Example 1: Example 2:

DI3. Factoring Factoring is like distribution in reverse. We are given an expression without parentheses and we have to rewrite it with parentheses, typically using the greatest common factor.

CT2. What does “combining” mean?Combining like terms means adding or subtracting them. Only like

terms may be combined. If a term cannot be combined, leave it the way it is. Example 1: Example 2:

Combining like terms / Distribution / Factoring

SD4. Converting units: Proportion strategySet up a proportion with the scale written as a fraction on one side. Make sure the units in the numerators match as well as the units in the denominators.

How wide is the building in real life?

SD3. Converting units: Multiplication strategy

Multiply the number given with the numbers in the scale written as a fraction. Make sure you write the fraction so that the units cancel.

How wide is the building in real life?

SD1. Scale factor The real life house will be how many times wider and taller than the blueprint? To make this comparison, we first need to make sure the units are the same. The key says every 2 cm on the blueprint corresponds to 45 ft in real life but 1 foot is about 30 centimeters. Therefore, 45 ft is about 45 × 30=1,350 cm. Now we can divide 1,350 cm by 2 cm to get 675, which is the scale factor. The width and height of the house will be about 675 times larger in real life than they appear on the blueprint.

Scale drawings

The questions that follow refer to the above picture.

Solving simple equationsSE1. Our goal is to get the unknown by itself.SE2. We get the unknown by itself by using inverse operations. Multiplication and division are inverses of each other and addition and subtraction are inverses of each other.SE3. The equal sign is what separates the two sides of an equation.SE4. Whenever an inverse operation is used on one side of an equation, the exact same thing must be done on the other side.SE5. To check to see if we got the right answer, we can take the value we got and plug it back into the original equation. If both sides are the same, our answer is correct.

2 cm

SD5. When finding area, you need to convert the length and the width

Example: What is the area of the face of the house (not including the roof) in real life?Solution:

6cm ×22.5 ftcm

=135 ft

2cm× 22.5 ftcm

=45 ft

135 ft × 45 ft=6,075 ft2

SD2. Unit scale To find the unit scale, divide both sides of the scale so that one of the numbers becomes 1.

2 cm=45 ft ÷ 2÷ 2 1 cm=22.5 ft

Solving complicated equationsCE1. Rules SE1-SE5 also apply to complicated equations.CE2. If a term is being added/subtracted while inside parentheses or a fraction, you cannot immediately get rid of it by using subtraction/addition.

Example 1: Incorrect Correct Correct 30=10(4 x+2) 30=10(4 x+2) 30=10(4 x+2) −2−2 10 10 28=10(4 x) 30=40 x+20 −20 −20 3=4 x+2 10=40 x −2−2 1=4 x

Example 2: Incorrect Correct Correct

7=30 x−105 7=30 x−10

5 7=30 x−105

+10+10 7=6 x−2 35=30 x−10

17=30 x5

+2+2 +10+10

9=6 x 45=30 x

CE3: If you choose to multiply or divide an equation, every term on both sides of the equation must get multiplied or divided. However, don’t forget to read CE4 below.

Example: Incorrect Correct Incorrect Correct

12−3 x=8 12−3 x=8 9+ x5=20 9+ x

5=20

3 3 3 3 ×5 × 5 45+ x=100

12−x=2 23 4−x=2 2

3 9+x=100

CE4: When dividing an equation, terms in parentheses stay the way they are. At first, you might think this goes against rule CE3 which says that every term gets divided. Actually we are not breaking rule CE3. What’s happening is that the terms are first getting multiplied by the number outside the parentheses and then getting divided; the operations cancel each other. The same is true when we multiply a fraction.

Example: Incorrect Correct Incorrect Correct

50=2(8−6 x) 50=2(8−6 x) 6=8 x+364 6=8 x+36

4 22 22

25=4−3 x 25=8−6x 24=32 x+144 24=8 x+36

because… 50=2(8−6 x) because… 6=8 x+364

50=16−12 x 6=2 x+9

Inequalities

IN1: What are inequality symbols?Inequality symbols are symbols that tell you what’s bigger and what’s smaller. Whatever is next to the “mouth” is bigger and whatever is next to the “point” is smaller.

IN2: The line underneathIf you see a line underneath the inequality symbol it should be read “or equal to” telling you that the two sides could be equal.

IN3: Reading inequalities You can read inequalities from left to right OR from right to left. 7<10Left to right: Seven is less than ten.Right to left: Ten is greater than seven.

9 ≥ 2

Left to right: Nine is greater than or equal to two.Right to left: Two is less than or equal to nine.

IN4: Solving inequalities Solving inequalities is exactly like solving equations with one extra rule. Whenever we multiply or divide both sides of an inequality by a negative number, we have to flip the inequality symbol. Briefly, this is because multiplying or dividing by a negative changes the signs so that the side that was bigger is now smaller and the side that was smaller is now bigger.

IN5: Graphing inequalitiesa. If the variable is next to the mouth, that means it’s bigger; shade to the right. If the variable is next to the point, that means it’s smaller; shade to the left. b. If there is a line underneath the inequality symbol, that means both sides could be equal to each other; fill in the circle.

Examples:

Similar figures

If two proportional shapes are given, you can find a missing side length by setting up a proportion. To figure out how to set up the proportion, it’s a good idea to draw arrows and put the number at the beginning of the arrow in the numerator and the number at the end of the arrow in the denominator. Arrows can be drawn within shapes or between them. Be careful! One of the shapes might be drawn upside-down, flipped around, etc.

SF1. Within shapes

Example: Each arrow stays within the same shape (small or big) and goes from hypotenuse to base.

First arrow: Small hypotenuse to small baseSecond arrow: Big hypotenuse to big base

SF2. Between shapes

Example: Each arrow connects the same side (hypotenuse or base) and goes from big to small.

First arrow: Small hypotenuse to big hypotenuseSecond arrow: Small base to big base

7 7

c 20

ProbabilityPB1. What is probability? How do you find a simple probability? Probability can be defined as the ratio of events that are considered “good” to the total number of possible events. It can also be defined as the chance of an event being successful. Based on the first definition, we can find the probability of an event by using the equation

P(event) = GoodTotal

.

PB2. What do probability values mean?P = 0: The event is impossible.

0¿ P<0.5 : The event probably won’t happen but it might.P = 0.5: The event is exactly as likely to happen as not to happen.

0.5<P<1: The event will probably happen but it might not.P=1: The event will definitely happen.

PB4. How do you find a simple probability with more than one good outcome If there is more than one outcome which is considered “good” then the probability of the event being successful is the sum of the probabilities of each good outcome.Example 1: Jessica will win a game if she spins the spinner and lands on 7 or 20. What is the probability of her winning?

Solution: P (7∨20 )=24+ 1

4=3

4Example 2: Victor will win a game if he reaches into the bag and pulls out a black, gray, or triangle marble. What is his probability of winning?

Solution: P (black∨gray∨triangle )=28+ 2

8+ 1

8=5

8

PB5a. How do you find a compound probability (more than one thing has to go right)? If you need more than one independent event to be successful then the probability of being successful overall is the product of the probability of success for each event.

Example: Jonathan will win a game if he spins the spinner and lands on c and pulls a white marble out of the bag. What is the probability of him winning?

Solution: P (c∧white )=14

× 38= 3

32

PB5b. How do you find the probability of something happening multiple times in a row? This is really just a compound probability for which each event happens to be the same. You would need to multiply the probability of the event with itself however many times in a row it is supposed to happen.

Example: Rajab will win a game if she spins the spinner and it lands on c five times in a row. What is the probability of her winning?Solution: P (c five׿a row ) ¿ P(c∧c∧c∧c∧c)

¿ 14

× 14

× 14

× 14

× 14= 1

1,024

PB7. Making predictionsStep 1: Find the probability (using P1, P4, or P5)Step 2: Multiply the probability with the number of attempts.

OR set up a proportion: Probability= xAttempts

Example: Michelle will win a penny every time the spinner lands on 20. If she spins the spinner 16 times, what is the best prediction you can make for how many pennies she will win?

Solution: Step 1: P(20)=14

OR

Step 2: 14

×16=4 P(20) =14

14= x

16 PB8. Representations

Example: Here’s how you could represent all possible outcomes from flipping a coin (heads or tails) and rolling a six-sided die (1, 2, 3, 4, 5, 6).

PB6. Be careful when things are not split up equally

The problems the follow refer to the spinner below which has four sections of equal sizeand the bag of marbles which has eight marbles of equal size.

Warning! Pay close attention to “or” and “and”. “Or” implies a simple event with more than one good outcome (you should add) while “and” implies a compound event (you should multiply). Notice that the probabilities in P4 all have “or” in them while the probabilities in P5a and P5b all have “and” in them. You can even have a probability with both “or” and “and”…

Example: Sven will win a game if he spins the spinner and it lands on 7 or c and he pulls a black or white marble out of the bag. What is the probability of him winning?Solution: P(7∨c∧black∨white)

¿( 24+ 1

4 )×( 28+3

8 )=34

× 58=15

32

For the spinner on the left…

What is P(A)? It’s not 14

!

What is P(B)? It’s not 14

!

PB3. Counting principle To find how many combinations there are, multiply the number of ways each thing can happen.

Example: I have 3 hats, 4 shirts, and 2 pairs of pants. How many different outfits can I make?

Solution: 3 × 4×2=24

A

B DC

A

B DC

A A

Type equation here .

PB8. RepresentationsExample: Here’s how you could represent all possible outcomes from flipping a coin (heads or tails) and rolling a six-sided die (1, 2, 3, 4, 5, 6).

PB6. Be careful when things are not split up equally

To find the probabilities, we have to imagine what the spinner would look like if it was split up into equal sections. Now we

can see that P( A)=36=1

2 and P (B )=1

6

.

CirclesCI1: Parts of a circle

Radius Diameter Circumference Area1-dimensional 1-dimensional 1-dimensional 2-dimensional

CI3: Going from radius to diameter and vice versaLook at the pictures of radius and diameter in P1. The diameter is twice as long as the radius. Therefore…

If you know the radius, multiply by 2 to get the diameter.If you know the diameter, divide by 2 to get the radius.

CI2: Pi (π )

Pi is the ratio of the circumference of a circle to its diameter. That is, if you divide the circumference of any circle by its diameter you always get the same number…

Cd

=π=¿

3.141592653589793238462643383….. It’s impossible for you to write this number all out because it never ends and never repeats. That’s why

CI4: The circle equations C=πd C=2πr A=π r2

where r is radius, d is diameter, C is circumference, and A is area.

CI5: Using the circle equations

Starting with the information you’re given, follow the paths, and use the equations or operations along those paths, to find anything else.

Example 1: If you know d and you want C, use C=πd.Example 2: If you know d and you want A, first divide by 2 to find r, then plug r into A=π r2.Example 3: If you know A and you want C, first use A=π r2 to find r, then plug r into C=2 πr .

CI6. Getting rid of the exponent in A=π r2

An exponent of 2 means we are squaring a number. That means multiplying the number with itself. The inverse of this is taking the square root (√❑ ). For example, √169 is 13 because 132 is 169. When you take the square root, ask yourself, “What number multiplied with itself will give me this number?”

Example: The area of a circle is 50 π . What is the circle’s radius?

Solution: A=π r2 169=r2

169π=π r2 √169=√r2

169 π=π r2 13=r π π

CI7: After you multiply by π, the π is not there anymore.

Example: The radius of a circle is 4 inches long. What is its circumference?

Solution: In terms of pi Without pi

CI8a: Cancel π ' s if there is a π in the numerator and the denominator. Don’t cancel it if there’s only one.

Example: The circumference of a circle is 18 inches long. What is the radius?

Incorrect Solution: This is WRONG!

CI8b: Use parentheses when dividing by 2 π in a calculator.Example: To answer the question above, put 18 ÷(2π ) into a calculator. The answer is ≈ 2.86. If you put 18÷ 2π in the calculator then the calculator will do 18 ÷ 2 first and then multiply by π and will give you an

This student cancelled the pi on the left even though there was no pi in the numerator to cancel it with.

CI7: After you multiply by π, the π is not there anymore.

Example: The radius of a circle is 4 inches long. What is its circumference?

Solution: In terms of pi Without pi

CI8a: Cancel π ' s if there is a π in the numerator and the denominator. Don’t cancel it if there’s only one.

Example: The circumference of a circle is 18 inches long. What is the radius?

Incorrect Solution: This is WRONG!

CI8b: Use parentheses when dividing by 2 π in a calculator.Example: To answer the question above, put 18 ÷(2π ) into a calculator. The answer is ≈ 2.86. If you put 18÷ 2π in the calculator then the calculator will do 18 ÷ 2 first and then multiply by π and will give you an

Measures of Center

Mean and median are called measures of center because they will give us a number that lies somewhere in the middle of the data set.

CS1. Mean (average): Add up all the numbers and then divide by how many numbers there are.

84+97+73+80+95=429

429÷ 5=85.8 Mean

CS2a. Median: First put the numbers in order from least to greatest! After that, simply find the middle number.

73 80 84 95 97

Median

CS2b. Finding the median when there are two middle numbers: If there are an even number of numbers, then there will be two middle numbers. If this happens, find the mean of the two middle numbers (add them up and divide by two). For example, if the number 88 were inserted into the data set above, here’s how we would calculate the median…

73 80 84 88 95 97

84+88=172

172÷2=86

In all of the examples below, except for the second median example and mode, the following data set was used… Data set: 84 97 73 80 95

70 75 80 85 90 95 100

Each dot on the number line represents a number in the data set.

The median is halfway between 84 and 88.

Visually Comparing Measures of Center and Spread

Measures of Center and Spread (and Mode)

When we are shown data plotted on a number line, it makes it easier to compare measures of center and spread just by using our eyes! On the dot plots below, the center of Class B’s test scores is clearly higher than that of Class A. However, the scores for Class A are more spread out. Therefore, the mean and median would be greater for Class B but the range and mean absolute deviation would be greater for Class A.

Measures of Spread

Range and mean absolute deviation are called measures of spread because they tell you how spread out the data is.

CS3. Range: Subtract the smallest number from the largest number.

97−73=24 Range

The largest number is 24 units away from the smallest number.

CS4. Mean absolute deviation: Find the mean of the data set. After that, find how far away each individual point of data is from the mean by taking the absolute value of the differences between them and the mean. After that, find the mean of those differences.

Step 1: The mean of the data set is 85.8 .

Step 2: |84−85.8|=1.8 |97−85.8|=11.2 |73−85.8|=12.8 |80−85.8|=5.8 |95−85.8|=9.2Step 3: 1.8+11.2+12.8+5.8+9.2=40.8 40.8 ÷ 5=8.16 Mean absolute deviation

Mode

CS5. Mode: Find the number that occurs the most.

Example: In the data set below, themode is 4 because that is the numberthat occurs the most.

9 4 59 42 4

On a dot plot, the mode would have the greatest number of dots above it.

CS6.

Measures of Center

Mean and median are called measures of center because they will give us a number that lies somewhere in the middle of the data set.

CS1. Mean (average): Add up all the numbers and then divide by how many numbers there are.

84+97+73+80+95=429

429÷ 5=85.8 Mean

CS2a. Median: First put the numbers in order from least to greatest! After that, simply find the middle number.

73 80 84 95 97

Median

CS2b. Finding the median when there are two middle numbers: If there are an even number of numbers, then there will be two middle numbers. If this happens, find the mean of the two middle numbers (add them up and divide by two). For example, if the number 88 were inserted into the data set above, here’s how we would calculate the median…

73 80 84 88 95 97

84+88=172

172÷2=86

The median is halfway between 84 and 88.

Visually Comparing Measures of Center and Spread

Measures of Spread

Range and mean absolute deviation are called measures of spread because they tell you how spread out the data is.

CS3. Range: Subtract the smallest number from the largest number.

97−73=24 Range

The largest number is 24 units away from the smallest number.

CS4. Mean absolute deviation: Find the mean of the data set. After that, find how far away each individual point of data is from the mean by taking the absolute value of the differences between them and the mean. After that, find the mean of those differences.

Step 1: The mean of the data set is 85.8 .

Step 2: |84−85.8|=1.8 |97−85.8|=11.2 |73−85.8|=12.8 |80−85.8|=5.8 |95−85.8|=9.2Step 3: 1.8+11.2+12.8+5.8+9.2=40.8 40.8 ÷ 5=8.16 Mean absolute deviation

Mode

CS5. Mode: Find the number that occurs the most.

Example: In the data set below, themode is 4 because that is the numberthat occurs the most.

9 4 59 42 4

On a dot plot, the mode would have the greatest number of dots above it.

SU2. Poor surveys: Not asking enough people

Question: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He asked Jamie, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?

Answer: No, the inferences would not be valid. To get a good idea of how students in class 711 feel, Mr. Frydman should ask a large number of them. It doesn’t have to be everyone, but certainly more than just one or even a few individuals.

SU3. Poor surveys: Survey bias

Question: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He randomly selected 20 students in the class and asked them, “Do you prefer noisy, smelly dogs or clean, cuddly cats?” Would the inferences that Mr. Frydman draws from his survey be valid?

Answer: No, the inferences would not be valid. There is bias within the survey itself because Mr. Frydman put his own opinion into the question. Calling dogs noisy and smelly and cats clean and cuddly would persuade more people to choose cats over dogs, even if that was not their own opinion before Mr. Frydman asked the question.

SU4. Poor surveys: Asking biased people

Question: Mr. Frydman wanted to know if the people in New York City generally preferred dogs or cats. He went to a dog park and asked 50 people, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?

Answer: No, the inferences would not be valid. People in a dog park are more likely than the general population to prefer dogs over cats. Their opinion would not be representative of people in New York City generally.

SU5. Valid surveys

Question: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He randomly selected 20 students in the class and asked them, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?

Answer: Yes, the inferences would be valid. Mr. Frydman chose the students randomly, he asked a large number of

SU1. Random sampling leads to valid inferences

Good: Mr. Frydman puts all of the students’ names into a hat. Each name is written on equal sized strips of paper. He then closes his eyes and begins pulling names out of the hat.Bad: Mr. Frydman puts all of the students’ names into a hat. The girls’ names are on larger strips of paper than the boys’ names. Mr. Frydman then closes his eyes and begins pulling names out of the hat. This is a bad sampling method because the girls in the class have a higher likelihood of getting picked than the boys.Good: Mr. Frydman assigns every student a number. He then uses a computer program to randomly generate numbers and chooses those students.Bad: Mr. Frydman assigns all the students on the checkers team a number. He then uses a computer program to randomly generate numbers and chooses those students. This is a bad sampling method because the students who are not on the checkers team have no chance of being picked.Good: Mr. Frydman asks all the students in the class to form a line. He then chooses every other person (second in line, fourth, sixth, eighth, and so on).Bad: Mr. Frydman arranges the students in a line so that they alternate girl boy girl boy. He then chooses every other person. This is a bad sampling method because the girls have no chance of being picked.

Surveys

Mr. Frydman wants to know if the students in class 711 prefer dogs or cats. The population (the group of interest) is class 711. Mr. Frydman wants to make sure his sample (the students he asks) is chosen randomly (Mr. Frydman should not be able to predict who will get chosen) and is representative of the whole class (conclusions drawn from the sample are similar to conclusions that would have been drawn had everyone been asked). This will ensure that the inferences (conclusions) that Mr. Frydman draws are valid (legitimate). What are some examples of good and bad sampling methods?

SU5. Valid surveys

Question: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He randomly selected 20 students in the class and asked them, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?

Answer: Yes, the inferences would be valid. Mr. Frydman chose the students randomly, he asked a large number of

Writing equations and inequalities

WE1. y=mx

Equations in this form show a proportional relationship between the variables x and y.

The constant m is the unit rate (constant of proportionality).

Example: Fiona studies 2 hours per day. If d represents the number of days that go by and t represents the total number of hours studied, write an equation that relates these two quantities.

Solution: Figure it out by thinking about what the total would be after a certain number of days.

After 0 days: t=0=2(0)After 1 day: t=2=2(1)After 2 days: t=2+2=2(2)After 3 days: t=2+2+2=2(3)After 97 days: t=2(97)After d days: t=2 d

WE5. Always check by plugging in For the example in WE1, what if a student is not sure if she should write d=2t or t=2d ? She can decide by thinking of an example and then plugging numbers into both equations and seeing which one makes sense. For instance, after 4 days Fiona will have studied a total of 8 hours. Let’s plug in 4 for d and 8 for t and see which equation makes sense…

d=2t t=2 d

4=2(8) 8=2(4 )

4=16 8=8 The first equation gives a nonsense result but the second equation checks out!

WE2. y=b+mx Equations in this form show a

linear relationship between the variables x and y but they are not proportional (on a graph we would see a straight line but it would not go through the origin unless b is 0).

m is the unit rate (but there is no constant of proportionality because the relationship between x and y is not proportional).

b is the initial value (starting point).

Example: Fiona has already studied for 6 hours. She will continue studying 2 hours per day. Write and equation relating t, the total number of hours studied, and d, the number of days studied after the original 6 hours.

Solution: Figure it out the same way as in WE1 but remember that Fiona started having already studied for 6 hours.

After 0 more days: t=6=6+2(0)After 1 day: t=6+2=6+2(1)After 2 days: t=6+2+2=6+2(2)After 3 days: t=6+2+2+2=6+2(3)After 584 days: t=6+2(584)After d days: t=6+2d

WE3a. y=q(x+r )

Equations in this form are also linear.

If we wanted to, we could write them like WE2 but writing them as above is easier for some problems.

Example: There are 6 buckets. Each bucket originally had d dandelions in it. 2 more dandelions were then added to each bucket. What is the total number, t, of dandelions in all buckets now?

Solution: Originally, there were 6 buckets and each bucket contained d dandelions…

Original total: t=6 dAfter two dandelions were added to each bucket, each of the 6 buckets contained d + 2 dandelions…

New total in all six buckets: t=6 (d+2)

WE3b. Parentheses are important When we write t=6 (d+2) we are saying there are 6 buckets and each bucket has d+2 dandelions in it. If we wrote t=6d+2 then we would be saying that each bucket only has d dandelions in it and there are an extra 2 dandelions somewhere…

This is incorrect for the problem. However, it is possible to write the correct equation without parentheses by distributing: t=6 (d+2) t=6d+12

In other words, there are an extra 12 dandelions total because we put an extra 2 in each bucket. By distributing, we can make a WE3a equation look more like a WE2 equation.

Warning: There could be negatives and minus signs! None of the examples on this page had minus signs or negatives but that was just to keep things simple. You will certainly have to include minus signs and negatives in some of your own equations and inequalities depending on the situation.

WE4. Writing inequalities When deciding which inequality symbol to use, try translating a phrase into another phrase with the words “less” or “more”. Pay close attention to when an inequality should have a line underneath.

Examples:

The number of students should be more than 10.No translation necessary: s>10The number of students should be no more than 10.Translation: …10 or less: s≤10The number of students should be less than 10.No translation necessary: s<10

The number of students should be no less than 10.Translation: …10 or more: s≥10The number of students should exceed 10.Translation: …more than 10: s>10The number of students should not exceed 10.Translation: …10 or less: s≤10The number of students is at least 10.Translation: …is 10 or more: s≥10The number of students should be at most 10.Translation: …10 or less: s≤10The minimum number of students is 10.Translation: The number is 10 or more: s≥10The maximum number of students is 10.Translation: The number is 10 or less: s≤10

d d d d d d

d+2 d+2 d+2 d+2 d+2 d+2