1. Expressions and Formulas What will we cover in this unit? creating formulas to solve problems Arrow strings Arithmetic trees Parentheses

2. Section A – Arrow Language Pay Attention to the Details Question # 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.

3. Page 2 - # 4 Number of Passengers Getting off the Bus Number of Passengers Getting on the Bus 58 913 16 158 93 Change 3 more 4 more No change 7 fewer 6 fewer 5 fewer

4. 10+6=16+3=19+3=22-4=18 Is 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.

5. Arrow Strings Arrow 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.

6. Ms. Moss page 2 - # 8 $1,235 - $357 $878 - $275 $603

7. 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?

8. Wandering Island Year Area Washed Away (km2) Area Added (km2) 19995.56.0 20006.03.5 20014.05.0 20026.57.5 20037.06.0 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

9. Section A Summary Arrow Language can be helpful to represent calculations Each calculation can be described with an arrow: starting number action resulting number A series of calculations can be described by an arrow string: 10 + 6 16 +3 19 +3 22 – 4 18

10. Section C - Formulas Tomatoes $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

11. Page 16, question 6 WeightTomatoes $1.20/lb Green Beans $0.80/lb Grapes $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

12. Taxi Fares We 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).

13. Taxi Fares We 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

14. Stacking Cups Rim Hold Base

15. Stacking Cups Two cups stackedFour cups stacked

16. How many cups will fit in this space? 50 cm Use an arrow string to find your answer.

17. 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.

18. Section C Summary A 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.

19. Section D – Reverse Operations Foreign 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

20. Convert Dollars to Guilders US Dollars Dutch Guilders 1 1.65 2 3.30 3 4.95 4 6.60 6 9.90 5 8.25 7 11.55 8 13.20 10 16.50 9 14.85

21. Estimation Marty’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.

22. Going Backwards Reverse 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 becomes 2 -4 6 10 60 +2 58 x2 29

23. Beech Trees Thickness: 20 x 0.4 8 -2.5 5.5 centimeters 30 x 0.4 12 -2.5 9.5 centimeters 40 x 0.4 16 -2.5 13.5 centimeters Height: 20 x 0.4 8 + 1 9 meters 30 x 0.4 12 + 1 13 meters 40 x 0.4 16 +1 17 meters

24. Section D Summary Every 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.

25. Order of Operations 1. Please - Parentheses 2. Excuse - Exponents 3. 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)

26. Order of Operations Parentheses 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

27. 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

28. 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

29. Practice Problems Copy 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 34. 3 + 8 – 7 – 2 x 2 5. 9 x 3 – 7 + 16. 18 – 6 x 2 + 1 x 7