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.
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
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
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