Expressions and FormulasWhat will we cover in this unit?

creating formulas to solve problems

Arrow strings

Arithmetic trees

Parentheses

Section A – Arrow LanguagePay Attention to the DetailsQuestion # 1 reminds us that we always have to pay attention to what the question is actually asking us, and not what we think the question is asking us.

Page 2 - # 4Number of

Passengers Getting off the Bus

Number of Passengers Getting

on the Bus

5 8

9 13

16 16

15 8

9 3

Change

3 more

4 more

No change

7 fewer

6 fewer

5 fewer

10+6=16+3=19+3=22-4=18Is this the correct way to write this?

No! Why not?

Because writing it this way means that 10+6 and 16+3 are equal.

10+6=16 and 16+3=19, they are not equal.

Therefore we cannot write them like we did above.

Arrow StringsArrow strings are formulas that we use to

help us organize addition, subtraction, multiplication, and division.

10+6=16+3=19+3=22-4=18 We know this is incorrect, but an arrow string can help us write it correctly.

10 + 6 16 +3 19 +3 22 – 4 18 This is the correct way to write this math problem. Since

it doesn’t have any equals signs, everything in our arrow string is true.

Ms. Moss page 2 - # 8

$1,235 - $357 $878 - $275 $603

Kate page 3, #9a and b

$37 +$10 $47 -$2 $45 +$5 $50 +$5 $55 -$2.75 $52.25 -$3 $49.25

Yes, she did have enough money to buy the radio on Wednesday.

What may have made that answer easier to find?

Wandering IslandYear

Area Washed

Away (km2)

Area Added (km2)

1999 5.5 6.0

2000 6.0 3.5

2001 4.0 5.0

2002 6.5 7.5

2003 7.0 6.0

Area of Island

(210 km2 to start)

210.5 km2

208 km2

209 km2

210 km2

209 km2

Change(km2)

+ 0.5 km

- 2.5 km

+ 1.0 km

+ 1.0 km

- 1.0 km

Section A SummaryArrow Language can be helpful to represent

calculationsEach calculation can be described with an

arrow: starting number action resulting

numberA series of calculations can be described by

an arrow string:

10 + 6 16 +3 19 +3 22 – 4 18

Section C - FormulasTomatoes $1.50/lb Price per pound

2.00 lb How many pounds were purchased

$3.00 Total price for 2 lbs of tomatoes

lb – is the abbreviation for pound

An arrow string can be created for this: Price per pound x number of pounds bought total cost

Page 16, question 6Weight Tomatoes

$1.20/lbGreen Beans

$0.80/lbGrapes $1.90/lb

0.5 lb

1.0 lb

2.0 lb

3.0 lb

$0.60

$1.20

$2.40

$3.60

$0.40

$0.80

$1.60

$2.40

$0.95

$1.90

$3.80

$5.70

Taxi FaresWe can set up these problems just like the

tomatoes.

4 miles number of miles traveled

$1.50 price per mile

$6.00 cost for travelling four miles

$2.00 one time fee that must be added to the cost

$8.00 Total cost (cost for travelling four miles + the one time fee).

Taxi FaresWe can create an arrow string for this

problem as well:

number of miles x price per mile cost for miles only

+ one time fee total cost

We can now substitute the words with the actual amounts:

4 x $1.50 $6.00 + $2.00 $8.00

Stacking Cups

Stacking CupsTwo cups stacked

Four cups stacked

How many cups will fit in this space?

50 cm

Use an arrow string to find your answer.

Are They the Same?Rule: Addition and subtraction can be done in

any order. Multiplication and division can be also

be done in any order. However, when addition or

subtraction is combined with multiplication or

division, the order of operations is important

because switching the order of operations

results in different answers.

Section C SummaryA formula shows a procedure that can be

used over and over again for different numbers in the same situation.

Sometimes it is possible to change the order of the arrows in a string. If a problem has only addition and subtraction or multiplication or division, the order can be changed, and the result will stay the same.

Section D – Reverse OperationsForeign Money

One United Stated dollar = 1.65 Dutch guilders

The arrow string to use for this formula is :

Number of Dollars x 1.65 number of guilders

Convert Dollars to Guilders

US Dollars

Dutch Guilders

11.65

23.30

34.95

46.60

69.90

58.25

711.5

5

813.2

0

1016.5

0

914.8

5

EstimationMarty’s Rule: Number of guilders 10 ____ x 6 number of

dollars

Will this work? YES. You divide by ten to see how many groups

of ten guilders there are. Then each one is six dollars, so multiply the number of groups by six.

Going BackwardsReverse strings are used when the answer to

the problem is known but the beginning of a problem is not known.

Always remember when you reverse your arrow string, you also have to reverse the operations in that arrow string.

For example:2 +4 6 x10 60 -2 58 2 29 becomes2 -4 6 10 60 +2 58 x2 29

Beech TreesThickness:20 x 0.4 8 -2.5 5.5 centimeters30 x 0.4 12 -2.5 9.5 centimeters40 x 0.4 16 -2.5 13.5 centimeters

Height: 20 x 0.4 8 + 1 9 meters30 x 0.4 12 + 1 13 meters40 x 0.4 16 +1 17 meters

Section D SummaryEvery arrow has a reverse arrow.

A reverse arrow has the opposite operation.

Reverse arrows can help us find the beginning number if all we know if the final number in a number string.

Order of Operations1. Please - Parentheses2. Excuse - Exponents3. My – Multiplication (or division, whichever

comes first in the number sentence)4. Dear – Division (or multiplication, whichever

comes first in the number sentence)5. Aunt – Addition (or subtraction, whichever

comes first in the number sentence)6. Sally – Subtraction (or addition, whichever

comes first in the number sentence)

Order of OperationsParentheses first –

3 + (4 x 2) 2

In this problem, you have to do 4 x 2 first since it is inside of the parentheses.

3 + (4 x 2) 2

8

Order of Operations The next step in the order of operations is

division for this problem. You have to divide eight by two since the division sign is between those two numbers.

3 + (4 x 2) 2

8 4

Order of Operations The final step for this problem is addition. You will

add three and four since they are the two numbers on each side of the addition sign.

3 + (4 x 2) 2

Arithmetic Trees: Used to

8 organize the order of

operations in your math

4 problems.

7

Practice ProblemsCopy these problems into your notebook and

complete them using the order of operations.

1. 4 x 6 – (3+4) 2. 15 – 3 x 4 + 2

3. 16 (5-1) x 3 4. 3 + 8 – 7 – 2 x 2

5. 9 x 3 – 7 + 1 6. 18 – 6 x 2 + 1 x 7