lesson 3.1 solving simple equations€¦ · lesson 3.1 – solving simple equations learning goal:...
TRANSCRIPT
Date: _____________________
Lesson 3.1 – Solving Simple Equations Learning goal: how to solve one-step and two-step equations using opposite operations.
WARM UP
Write three more examples of an expression and three more examples of an equation.
Expressions Equations 5x – 7
9x – 8 + 2x
4(x – 6)
3x + 2 = 5
3x – 4 = 8x – 9
8(3x – 1) = 32
Compare expressions to equations. State what is similar and what is different.
Fill in the blanks with a number that makes the equation true:
a) 3 + ____ = 8 b) 5 – ____ = 1 c) 15 + _____ = 10
d) –5 + ____ = -7 e) – 3 + ____ = 8 f) 7 – _____ = 13
g) 4 x _____ = 24 h) _____ ÷ -2 = 12 i) 5 ( ____ ) = -30
j ) 12 ____ = -3 k) _____ 4 = -2 l) ____ (-9) = 4
m ) (___ 2) + 7 = 13 n) (____)(9) - 4 = 32 o) 5 ( _____ ) - 6 = 44
Explain your strategy for solving question m) above
KEEPING THE BALANCE
We can use a balance scale to model an equation.
When the pans are balanced the mass in one pan is equal to that of the other.
Date: _____________________
5g 5g 5g 5g 10g
5
g 5g 5g 5
g
10g
5g 5g 10g
For example, the pans below are balanced because we equal amounts on each side.
=
We can add or subtract any mass to this scale without upsetting the balance as long as we: This is called the BALANCE PRINCIPLE Example 1: Lets remove 5g from both sides of this scale.
=
Notice that the scale remains balanced because we removed the same amount from each side. We still have 10g in each pan.
=
Date: _____________________
X 5g
X 12g 17g
X 12g 17g
-12 -12
We can treat an equation like a balance with an unknown mass on one side (the variable). By examining the known values we can determine the missing mass. Example 2: Look at the balance below…
=
Write an equation to represent the balance above: If we remove 12 from the left pan then x will be left alone. But if we remove 12 from the left pan we must remove 12 from the right pan to keep the scale balanced.
=
=
So 𝒙 =
Date: _____________________
SOLVING EQUATIONS USING OPPOSITE OPERATIONS Addition is opposite to , and Multiplication is opposite to
Subtraction is opposite to , and Division is opposite to
Example 3: Solve the following equations using opposite operations.
a) 𝑎 − 8 = 12
What is being done to a?
What operation can undo this?
Isolate for a.
b) 5 + 𝑘 = −10
What is being done to k?
What operation can undo this?
Isolate for k.
c) 8𝑎 = 16
What is being done to a?
What operation can undo this?
Isolate for a.
d) 𝑥
3= 4
What is being done to x?
What operation can undo this?
Isolate for x.
Success Criteria for Solving One-Step Equations
1.
2.
3.
Date: _____________________
Example 4: Solve the following equations using opposite operations a) 𝑝 − 15 = −5 b) 2𝑏 = 20 c) 𝑡 + 8 = 12 d)
𝑥
6= 6
When we need to do multiple operations to isolate 𝑥 we use…
Example 5: Solve each of the following equations.
a) −2𝑦 − 3 = 7
What is being done to y?
What operations can undo this?
Isolate for a.
b) −17 =𝑥
9+ 15
What is being done to x?
What operations can undo this?
Isolate for x.
CHECK:
CHECK:
e) 12 = −𝑡 + 7
What is being done to t?
What operation can undo this?
Isolate for t.
f) 2𝑓
5+ 4 = 10
What is being done to f?
What operation can undo this?
Isolate for f.
Date: _____________________
Example 6: If I asked you to solve the equation x2 = 25,
a. What is the inverse operation of squaring a number?
b. What is a solution?
c. Is there a second solution that works?
Lesson 3.2 - Solving Equations with the Distributive Property Learning goal: By the end of this lesson I will demonstrate how to solve equations involving the distributive property. Recall:
• When solving an expression, we use the acronym ___________________ to remember the order of operations.
• When isolating for a variable we use the acronym _________________ to remember the order of operations.
Example 1: Solve the following equations using opposite operations a) 𝑥 + 8 = −4 b)
𝑏
4− 7 = −9 c) −3𝑡 + 20 = 5 d) 18 −
3𝑥
4= 24
Success Criteria for Solving Multi-Step Equations
1.
2.
3.
Date: _____________________
Example 2: For each word problem write an equation that models the situation, then solve. Remember to define the variables with a “let” statement first.
a) Colin ordered 3 pizzas. When he went to the store to pick up the pizzas it came to $27.00. How much did it cost for each pizza?
b) If Colin ordered 3 pizzas but paid a $1.50 charge for delivery how much was each pizza? (Assume the total is still $27.00)
c) A banquet hall charged $200 for the rental of the hall, plus $24 for each meal served. The total bill for the banquet was $3800. How many people attended the banquet?
d) Three more than twice a number is fifteen. Find the number.
Date: _____________________
SOLVING WITH THE DISTRIBUTIVE PROPERTY When you have brackets in the equation you need to expand and simplify before you can solve. Example 3: Solve.
a) 3(𝑘 + 8) = 21 b) 10 − 2(𝑤 − 4) = 12 c) 28 = 9𝑚 − 2(2𝑚 + 6) d) 2(3𝑥 + 5) + 3(2𝑥 + 5) = 1
Example 4: A rectangle has a length of 7x – 4 and a width of 8 – 2x. The perimeter of the rectangle is 38 cm. Determine the length of each of the sides.
Success Criteria for Solving Multi-Step Equations (Involving Distributive Property)
4.
5.
6.
Date: _____________________
Example 5: The figure below has a perimeter of 77 m. Determine the length of side AB. Show your work.
Example 6: Is 𝑥 = −1 a solution to the following equation?
3𝑥 + 2(𝑥 − 6) = −5(𝑥 + 4)
Date: _____________________
Lesson 3.3 – Solving Multi-Step Equations Learning Goal: By the end of this lesson I will demonstrate how to solve equations with variables on both sides. WARM UP Example 1: Solve the following equations using opposite operations. Do a formal check on c). a) 4– 2𝑥 + 6 – 3 = − 5 b) 5(𝑥 + 4) = 40 – (−5) c) 7𝑥 – 9𝑥 – 6 = 21 – 5 When you have variables on both sides of the equal signs, you must get all the variables to one side of the equal sign before you can isolate. Example 2: Solve and complete formal checks on a) .
a) 2𝑦 + 5𝑦 = 14 − 2 + 4𝑦 b) 𝑤 + 7 − 5𝑤 = −3 − 14𝑤 c) 𝑥 − (−3𝑥 + 5) = 7(𝑥 − 1) + 3
Example 4: The Sun Spa charges annual dues of $123 plus $10 per hour to use the facilities. The Moon Spa charges annual dues of $230 plus $7 per hour to use the facilities. For what number of hours would the two spas charge the same total amount?
Success Criteria for Solving Equations
7.
8.
9.
10.
Date: _____________________
Exmaple 5: A square and an equilateral triangle are pictured below. The square and the triangle have the same perimeter. What is the value of x?
Example 6: The perimeter of a rectangle is 30cm. If the length is 3 cm longer than the width, find the rectangle’s dimensions.
Date: _____________________
Lesson 3.4 – Solving Equations with Fractions Learning goal: By the end of this lesson I will demonstrate how to find a common denominator to solve equations with fractions.
WARM UP Example 1: Solve the following equations using opposite operations. a) 7𝑘 − 4 = 3𝑘 + 16 + 2𝑘
b) 7𝑛 − 3 = 9(𝑛 + 3)
c) 2(9𝑥 − 1) = 99 − 7(3 − 4𝑥)
If there is only ONE fraction use SAMDEB to solve normally. Example 2: Solve each of the following equations.
a) −1
4=
𝑦
3 b)
𝑣+8
5= 5 c)
2𝑥
3+ 1 = −4 d)
2(𝑒+5)
3= −2
When solving equations involving multiple fractions, multiply both sides by the lowest common denominator to first to eliminate all fractions. Example 3: Solve each of the following equations.
a) 4𝑦
3−
1
2= 4 LCD: __________ b)
𝑏+3
4=
𝑏−1
2 LCD: __________
c) 𝑥−3
6−
𝑥−25
5= 4 LCD: __________ d)
𝑥+5
2−
𝑥−1
2=
𝑥−1
3 LCD: __________
Example 3: One third of a number is 15 less than five sixths of a number. Find the number.
Success Criteria for Solving Equations
1.
2.
3.
4.
Date: _____________________
Example 4: Lauren spends one fourth of her money on food, three tenths of her money on make-up, and two fifths of her money on clothes. If she spent $95 on these three items all together, how much money did she have to begin with?
Lesson 3.5 – Working with Equations and Formulas Learning Goal: By the end of this lesson I will communicate statements that define variables and demonstrate how to rearrange a formula to isolate a variable.
Example 1: The following formula can be used to convert temperatures from degrees Celsius, °C, to degrees Fahrenheit, °F:
.
a) Use the formula to convert 20°𝐶 to F. b) Use the formula to convert 85𝐹 to °𝐶. REARRANGING FORMULAS When working with math, (whether in Chemistry class, Physics class, Business class, etc), you will be working with equations that will need to be rearranged. This makes them easier to work with. Rearranging equations is a very important skill in mathematics and follows the same rules as solving equations. Example 2: Rearrange each formula to isolate the variable indicated.
a)
isolate for h b) 2y + 3x = 30 isolate for y
c) isolate for h d) V = r 2h isolate for r
STANDARD FORM Since equations can be rearranged in a number of different ways, mathematicians have agreed upon a set of rules known as standard form of an equation. Equations in standard form must….
5
1609 +=
CF
2
bhA =
SA = 2pr2 +2prh
Date: _____________________
Example 3: Rearrange each equation to standard form. a) 𝑦 = −3𝑥 + 7 b) 9𝑥 − 8𝑧 = −3𝑦 + 10
c) 𝑦 = −4
5𝑥 + 3 d)
2
3𝑥 +
1
5𝑦 = 2
Lesson 3.6 – Solving Word Problems Learning goal: By the end of this lesson I will extract key information from written statements to set up and solve an equation. I will also effectively communicate my findings with concluding statements.
Tips to help you write your “let” & “then” statements:
→ Consecutive numbers would be _______________________________________________
→ Consecutive even/odd numbers would be: _______________________________________
→ When you are given the sum of two numbers, you can write one number as a difference.
Example #1: The sum of two numbers is 20. Let 𝑥 be one of the numbers. Then, _______________ is the other number.
Example #2: Tyrone has 52 jellybeans that are red and green. _______________________________________
________________________________________________________________________________________________________________.
Example 3: Quin was shopping at a used book sale where all books were selling at the same price. He bought six science fiction books and eight mysteries. He also decided to buy a poster for $2.40. In total, Quin spent $8.70. What was the price of a single book? Example 4: Oberon Cell Phone Company advertises service for 3 cents per minute plus a monthly fee of $29.95. If Parker’s phone bill for October was $38.95, find the number of minutes he used.
Date: _____________________
Example 5: Rachael and Sabine belong to different local gyms. Rachael pays $35 per month and a one-time registration fee of $15. Sabine pays only $25 per month but had to pay a $75 registration fee. After how many months will Rachael and Sabine have spent the same amount on their gym memberships? Example 6: Ella has an older sister and a younger sister. Her older sister is one year more than twice Ella’s age. Ella’s younger sister is three years younger than she is. The sum of their three ages is 26. Find Ella’s age. Example 7: Find two consecutive integers such that their sum is 89.
Date: _____________________
Example 8: Find three consecutive even integers such that the sum of twice the first and three times the third is fourteen more than four times the second.
Example 9: Patrick has 35 coins (only nickels and dimes) worth a total of $3.25. How many of each coin does he have?
Date: _____________________
Lesson 3.7 – Solving Ratios and Proportions Learning goal: By the end of this class I will demonstrate how to apply knowledge of solving equations to ratios and proportions.
WARM UP
Ratio – a comparison of two numbers by divison. Four out of five cars were red. Write this ratio in three different ways: Rate – a ratio of two measurments with different units. Mr. Underwood ran 400 yards in 80 seconds. Write this as a rate Unit Rate – a rate in which the denominator is 1 In example above, write Mr. Underwood’s unit rate
Jill is selling cookie dough. Three tubs cost $22.50. a) What is the unit rate? b) How much will 8 tubs cost?
A basketball player made 12 free throws in 4 games this year. About how many free throws would you expect the player to make in 100 games?
Proportion – an equation that shows that two ratios are equivalent. Fred got two out of every three questions right. If there were six questions, Fred got four questions correct.
3
2 =
6
4 is a proportion Note: Reciprocal of proportions are also equal
Date: _____________________
SOLVING PROPORTIONS Example 1: Solve each proportion.
a) 𝑥
5=
4
7 b)
6
7=
4
𝑥 c)
2
5=
6
𝑥+1
Example 2: An employee making $28,000 receives a raise of $1000. All other employees in the company are given proportional raises. How much of a raise would an employee making $32,000 receive? Example 3: A train travels at a steady speed. It covers 15 miles in 20 minutes. How far will it travel in 7 hours, assuming it continues at the same rate? PART TO TOTAL PROBLEMS Example 4: The ratio of red marbles to blue marbles is 5 to 7. If there are 156 marbles total, how many red marbles are there?
Date: _____________________
Example 5: Carl, Rani and Katy win a prize of $600. They decide to share the prize in the ratio of their ages. Carl is 15, Rani is 10 and Katy is 5. How much does each of them get?
Example 6: Ms. Marsh’s class is making lemonade to sell for a fundraiser. It contains a lemon juice and water in ratio the ratio of 5:2.
a) How many mLs of lemon juice are there for 250mL of water?
b) How many mLs of water are there for 750mL of lemon juice?
c) How many mLs of each liquid do you need for 150mL of lemonade?
Date: _____________________
Lesson 3.8 – Solving Percentage Problems Learning goal: By the end of this lesson I will demonstrate how to solve word problems involving percentages.
WARM UP
What does the word percent mean?
When is percent used outside of math class?
Complete the table below:
REDUCED FRACTION
DECIMAL PERCENT
155%
9
16
0.06
43.2%
In her math class, Samantha’s two most recent quiz grades are 29/35 and 22/26. Which quiz grade represents the better score?
PERCENT REVIEW A percent is a ratio between two quantities in which the second quantity is always 100.
84% means 84
100 or 0.84.
You need to have a solid working knowledge of percents, not only for success in this course, but also because percents are used in so many areas in our lives outside of school, such as sports statistics, taxes, wage increases, and sales commissions. Because finding a percent involves work with equivalent fractions, many percent questions are easily handled by solving a proportion of the following type:
part
total=percent
100
is
of=
%
100
Note: It may be helpful to think of the ratio “part/total” as “is/of.” The whole is often preceded by the word “of” in the problem. The part (or percentage) often precedes the word “is” in the problem.
Date: _____________________
Example 1: Answer each of the following questions by setting up an appropriate proportion and solving for the unknown. Round answers to the nearest tenth, where appropriate. a) What number is 18% of 200?
b) 42 is what percent of 98?
c) 6% of 240,000 is what number?
d) 12% of what number is 1044?
Answer each of the following exercises by setting up and solving an appropriate proportion. Example 2: Tanisha is about to sell her house for $315,000. Her real estate broker will expect to receive a 6% commission for all of her hard work in finding Tanisha a buyer for her house. How much money will the real estate broker expect to be paid once Tanisha’s home is sold? Example 3: Quinn decided to raise all of the prices in his store by 4%. What is the new price of an $8.25 item after the 4% increase? What is the final cost after 13% sales tax is added on? Example 4: Discount Dave’s is having an end of summer clearance sale. All items are 60% off. What is the sale price of a flat screen TV normally priced at $3499? What is the final cost after 13% sales tax is added on?
Date: _____________________
Example 5: Harleen is shopping at the mall. She notices a shirt for 40% off. How much did the shirt cost if she received a $17.68 discount? Example 5: Sheila claims that if you increase a number by 10% and then decrease the result by 10%, you are right back where you started. Show that Sheila is incorrect by using a specific numerical example.