Web viewPowerTeaching Math 3rd Edition© 2014 Success for All FoundationAlgebra 1 Unit 3 Cycle 1 Lesson 1Homework Problems1. PowerTeaching Math 3rd Edition

Homework Problems

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Write the vocabulary introduced in this cycle:

Today we created and solved equations and inequalities in one variable. Here’s an example!

The difference of 5 as a factor b times and 42 is 83.

We can write an equation for the situation: 5b – 42 = 83

Solution: 5b – 42 = 83

+ 42 = + 42

5b = 125

b = 3

1) Elisa checked out a library book and forgot to return it on time. There is an initial late fee of $2 and a fee of t cents owed for each day a book is late. Elisa returned her book 3 days late and was charged $2.75.

Write an equation or inequality for the situation. Then find how many cents are charged each day a book is late.

PowerTeaching Math 3rd Edition© 2014 Success for All Foundation

Algebra 1 Unit 3 Cycle 1 Lesson 1Homework Problems2

2) 12 plus 2 as a factor t times equals 16.

Write an equation or inequality for the situation. Then solve for t.

3) Jaheim is working with the Spring Dance committee to raise money for a spring dance at his school.They are selling potted tulips. The committee spent $250 on supplies and plans to sell each pot of tulips for $5 each. Their goal is to raise at least $800.

Write an equation or inequality for the situation. Then find how many pots of tulips must be sold for the committee to meet their goal. Explain your thinking.

4) Siobhan measured a cube that has side lengths of 5 inches. She used 5 inches as a factor q times to calculate the measurement 125 inches.

Write an equation or inequality for the situation. Then find what measurement Siobhan calculated.

5) The sum of 6 and the product of 3 multiplied by m is less than or equal to 18.

Write an equation or inequality for the situation. Then solve for m.

Mixed Practice

6) Identify the coefficient, factors, and base.

– 1 ac4

3

a. coefficient

b. factors

c. What is the base for the exponent 4?

7) Write two different math statements to describe the algebraic expression x2 – y2.

8)a. Give an appropriate measure for the

temperature.

b. What is the possible range of measurements of the temperature?

9) Libby deposited $930 in a savings account that pays interest at an annual rate of 0.75%. How much interest will she earn in 1 year?

Word Problem

10) Lucas purchased 4 large posters and 6 small posters online and spent $21 on shipping. He spent x to ship each small poster and twice as much to ship each large poster. What is the cost of shipping a small poster? Explain your thinking.



For the Guide on the Side

Today your student wrote equations and inequalities for situations that had one variable. He or she compared equations in one variable to expressions in one variable and saw that you can solve for the variable in equations. This is because an equation has two expressions that are equal to each other, or have the same value. Similarly, you can also solve for the variable in inequalities because an inequality is a comparison of two expressions. When your student solved for the variables in this lesson, he or shesaw that there was one value for the variable in equations. In the inequalities, there was a solution set such as x > 43 or x ≤ 3. In the next lesson, your student will expand upon these concepts and create and solve equations that have two or more variables.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about cr eat i ng eq u at i ons and i n e qu a l i t i es i n one v a ri a b l e:

1) What’s going on in this situation?

2) How can you tell if the situation represents an equation or inequality?

3) Do you think there will be more than one solution to this problem?

He r e a r e s o m e i d eas to w o r k w i t h cr ea t i ng e q ua t i ons a nd i ne q u a li t i es i n o ne v a r i a b l e:

1) What are some situations in your life that you can use an equation or inequality to describe? Is there more than one solution to your problem?

2) Watch a video on Khan Academy:

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / i ne q u a li t i e s / v / i ne q ua l i t i es

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / i ne q u a li t i e s / v / m u l t i-s tep- i n e qu a li t i es

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / v /o n e -s tep -i ne q u a l i t i es

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / v / s o l v i n g _ i n eq u a l i t i es

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a/ s o l v i n g - l i nea r- e q ua t i on s- a n d- i n e qu a li t i e s / e q u at i on s _ b eg i nne r / v / s o lv i ng - o ne -s tep - eq uat i ons

Homework Answers

1) 2 + 3t = 2.75$0.25 are charged each day a book is late.

2) 12 + 2t = 16t = 2

3) 5y – 250 ≥ 800The committee needs to sell 210 or more pots of tulips to meet their goal.Possible explanation: I translated this situation into math as $5 times y pots minus $250 for supplies is greater than or equal to $800. I isolated y on one side of the inequality to find the value that y hasto be greater than or equal to. That is 210.

4) 5q = 125Siobhan found the volume of the cube.

5) 6 + 3m ≤ 18m ≤ 4

Mixed Practice

6) a. coefficient: – 13

b. –1 •

c. c

1 • a • c • c • c • c or –1 •3

1 • a • c4

3

7) Possible answers: the difference of two squares; y squared less than x squared

8) a. –20°Cb. –20.5°C – –19.5°C

9) She will earn $6.98 in one year.

Word Problem

10) Regular shipping costs $1.50 per small posterPossible explanation: First, I wrote an equation for the situation. 6x + 4(2x) = 21. I knew that the sum of all shipping costs was 21. x is the cost to ship a small poster, so 2x is the cost to ship a largeposter. I then solved for x to find the cost of shipping one small poster.

Homework Problems

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Vocabulary words introduced in this cycle:

equation, inequality, x-intercept, y-intercept, exponential equation, constraint

Today we wrote equations that had two or more variables. We also graphed equations with two variables. Here’s an example!

Lin is printing a photo book. Each book costs $15 and contains 20 pages. It costs an additional$1.50 for each extra page added.

We can write an equation for the situation:y = total costx = number of additional pagesy = 15 + 1.5x

Use the graph to see possible total costs for different amounts of extra pages added.

x

0

5 22.5

10

If there are 0 additional pages, the book costs $15.

This relationship is linear because the cost increases at a constant rate. For each additional page, the cost increases by $1.50.



1) Celia has to call a plumber. Pierre’s Plumbing Company charges $50 for a service call and $75 for each hour of work done. Paolo’s Plumbers charges $50 for a service call and $90 for each hour of work done.

a. Write an equation for the total cost of work done by each plumbing company.

b. Create one graph for each equation on graph paper.

c. Compare and contrast the graphs for each plumbing company’s work charge.

2) b is equal to 430 increased by 4 to the power of a

a. Write an equation for the situation.

b. Create a graph for this equation on graph paper.

3) Luke is helping to fill up boxes for a canned-food drive. He has some total number of cans of food and he can fit 25 cans into each box that he fills. Luke has already placed 325 cans into boxes.

a. Write an equation to find the total number of cans that Luke has to box.

b. Create a graph for this equation on graph paper.

4) Melinda creates and sells candles and candle holders. She charges $6.50 for each candle and $2 for each candle holder. Write an equation to represent Melinda’s sales in one week.

Mixed Practice

5) Find the value of x for:

19 is the sum of 3 multiplied by itself x times and –8.

6) How many terms are in the expression 3 + y + 2xy – xy2?

7) Solve. Round to the correct number of significant figures.

453 cm • 0.32 cm

8) Simplify.

x(x2 + 4x + x)

Word Problem

9) Leah said she could graph the expression 2x + 3 by turning it into the equation 2x + 3 = y. Is she correct? Explain your thinking.




Today your student created linear and exponential equations that had two or more variables. A linear equation grows at a constant rate (for example if you babysit, you make $7 (+7) for every hour of babysitting). An exponential equation grows at a constant factor (for example a population of bacteria doubles (×2) every hour).

Your student also graphed linear equations with two variables. He or she compared graphs of similar equations to explore what the y-intercept (point at which the line crosses the y-axis) and slope (the slant of the line) tells you about the situation. In the next lesson, your student will look at similar equations and explore how to interpret mathematical solutions for real-world contexts.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about e q ua t i ons i n t wo or m o r e v a r i a b l e s : 1) How many variables will the equation have? How do you know?2) What variable will you graph on the x-axis? Why?3) Is this increasing/decreasing at a constant rate? How do you know?4) Is this a linear or exponential equation? How do you know?5) What are the x- and y-intercepts?6) Explain what you think the graph of this equation will look like.

He r e a r e s o m e i d eas to w o r k w i t h e qua t i o ns i n t w o or m o r e v a r i a b l e s :

1) What are some situations in your life that you can use an equation to describe? Can you write and graph the equation?


http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g r e b r a/ l i n ea r- e q uat i on s- a n d- i n e qu a li t i e / g r a p h i n g_ s o l u t i o n s 2/ v / g r aph s- o f - l i nea r- e q u a t i ons

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g r e b r a/ l i n ea r- e q uat i on s- a n d- i n e qu a li t i e / g r a p h i n g _ w i t h _ i nte rc ept s / v /g r a ph i ng - u si ng - x- and - y -i nte rc epts

http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r- e q uat i on s- a n d - i n eq u a l i t i e/e qu at i o n - o f - a- li n e / v /g r a ph i ng - a -l i n e - i n -sl o pe - i n t e rc ept - f o r m ? v= u k 7g S 3 cZ V p4

http s :/ / www. k hana c a d e m y . o r g/ m ath/t r i g o no m et r y / e x p o nen t i a l _ a nd _ l o ga r i t h m i c - f un c /e x p_g r o w t h_ d e c a y / v /g r aph i ng - e x p o n e nt i a l - f un c t i o ns

http s :/ / www. k hana c a d e m y . o r g/ m ath/t r i g o no m et r y / e x p o nen t i a l _ a nd _ l o ga r i t h m i c _ f u n c /e x p_g r o wth_ de c a y / v / e x po n en t i a l - g r o w th - f un c t i ons

Homework Answers

1) a. Pierre’s: y = 50 + 75x; y = total cost of work; x = hours workedPaolo’s: y = 50 + 90x; y = total cost of work; x = hours worked

b.

c. The y-intercepts are the same on both graphs because both companies charge the same flat fee to come to your home. However, Paolo’s Plumbers charges a higher rate per hour, so its line on the graph is steeper. Therefore, the graphs start at the same point but are not parallel.

2) a. b = 430 + (4)a

b.



3) a. c = 325 + 25b; c = total number of cans and b = the number of boxes Luke needs b.

4) c = number of candles sold in one weekh = number of candle holders sold in one weekx = sales in one weekx = 6.5c + 2h

Mixed Practice

5) x = 3

6) 4 terms

7) 1.4 × 102 cm2

8) x3 + 5x2

Word Problem

9) Possible explanation: Yes, Leah is correct. y is equal to the value of the expression, so y depends on the value of x. Leah could graph this equation.

Homework Problems

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Today we explored constraints for equations and inequalities. Here’s an example!

Salvator purchased lights for his new business.

On the first day, he purchased 3 LED light bulbs and 8 halogen light bulbs for a total of 52 dollars.

On the second day, he purchased 9 LED light bulbs and 12 halogen ones for a total of 96 dollars.

We can write a set of equations to describe this situation.d = cost, in dollars, of LEDh = cost, in dollars, of halogenfirst day3d + 8h = 52second day9d + 12h = 96Sherri thinks LED light bulbs cost $8 and halogen light bulbs cost $2. Is she correct?

No, Sherri is not correct because her prices do not make sense for both equations.first day3(8) + 8(2) ≠ 52second day9(8) + 12(2) = 96

PowerTeaching 3rd Edition© 2014 Success for All Foundation

Algebra 1 Unit 3 Cycle 1 Lesson 3Homework Problems 1

1) Jill is ordering fabric for a project. Yellow fabric costs $10 per yard and green fabric costs $7 per yard.She has a budget of $40 and needs at least 3 yards of fabric. She also knows she needs as much or more yellow fabric as green fabric. Write a set of equations and/or inequalities to represent this situation. Then explain whether Jill could order 2 yards of green fabric and 3 yards of yellow fabric.Explain your thinking.

2) Bryant bought q bags of chips for a party that cost $3 each. He used a $2-off coupon and paidr dollars in all. Write an equation or inequality to represent the situation. Then explain whether r could equal 16.

3) Yellow Cab Taxi Service charges a $2.50 flat rate and an additional $0.65 per mile. After looking at his budget, Roberto cannot spend any more than $10 on a ride. Write an equation or inequality to represent the situation. Then explain whether Roberto could travel 14 miles in a taxi.

4) Penelope has x 32-cent stamps, y 29-cent stamps, and z 3-cent stamps. The number of 29-cent stamps is 9 less than the number of 32-cent stamps, while the number of 3-cent stamps is 4 less than the number of 29-cent stamps. The total value of the stamps is $11.08. Write a set of equationsand/or inequalities to represent this situation. Then explain whether the following equations could be true: x = 22, y = 13, and z = 9.

Mixed Practice

5) Write an inequality for the situation: n is at least the sum the ratio of 4 and 5 and 10.

6) The diameter of the nucleus of a gold atom is 1.4 × 10–14 meters and the diameter of the nucleus of an aluminum atom is 7.2 × 10–15 meters. How many times smaller is the diameter of an aluminum atom than a gold atom?

7) A car gets 36 miles per gallon. If gas costs $3.88 per gallon, then how much will it cost to travel3,670 miles?

8) Find the area. Round to the correct number of significant figures. The formula for area of circle isA = πr2.

Word Problem

9) Explain why a solution to a set of inequalities and/or equations must satisfy all the inequalities and/or equations.

PowerTeaching 3rd Edition© 2014 Success for All Foundation



Today your student wrote multiple equations and/or inequalities to represent different conditions or constraints for the same situation. He or she identified whether a given value was a solution to the problem by determining whether it made every inequality and/or equation true. Your student also determined if a mathematically correct solution made sense in the context of the problem. For example, an equation may have both positive and negative solutions; however, if the equation represents measurements, it would only make sense to have positive number measurements for solutions. This skill is important for your student to practice as he or she will get into more complex math that represents real- world situations.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about c on s t r a i nt s :

1) What’s going on in this problem?

2) What are the variables?

3) How do you know if you need an inequality or equation to describe this situation?

4) What solutions will make sense? How do you know?

He r e a r e s o m e i d eas to w o r k w i t h c on s t r a i nt s :

1) What are some situations in your life that you can use an equation or inequality to describe? Are there any constraints? What type of solutions would make sense? For example, could your solution be negative?

2) Watch a video on writing systems of equations and inequalities:http:/ / ww . vi r t u a l ne r d. c o m /a l geb r a - 1/ s y s te m s- equat i o n s- i ne q u a li t i e s / i ne q u a li t i e s- d ef i n i t i on. p hp

3) Watch a video on inequality solutions:http:/ / www . vi r t ua l ne r d. c o m /p r e - a l g eb r a / l i nea r- f un c t i on s - g r aph i ng / i n e q u a l i t y -c h e c k - o r de r ed - p a i r - s o l u t i on . php

Homework Answers

1) 10y + 7g ≤ 40y + g ≥ 3y ≥ gNo, she could not order those amounts.Possible explanation: Jill could not order 2 yards of green fabric and 3 yards of yellow fabric.When I substituted the numbers in the inequalities, the values satisfied the inequality y ≥ g and the inequality y + g ≥ 3. But, the values did not satisfy 10y + 7g ≤ 40 because it resulted in44 ≤ 40 which is not true. I built a math model that incorporated the information given and the constraints in the situation. Then I plugged in 2 yards for the green fabric and 3 yards for the yellow fabric to see if the inequalities remained true.

2) 3q – 2 = rYes, r could equal 16.Possible explanation: To determine if r = 16 could be a solution, I solved for q when r = 16 and got 6. The variable q represents the number of bags of chips, and since I got a whole number,this made sense.

3) 2.5 + 0.65m ≤ 10No, he could not travel that far.Possible explanation: Roberto could not travel 14 miles in a taxi. To spend $10 or less, he canonly travel up to 11.5 miles.

4) 0.32x + 0.29y + 0.03z = 11.08y = x – 9z = y – 4Yes, those equations could be true.Possible explanation: These values satisfy all three equations in the system.

Mixed Practice

5) n ≥ 4 + 105

6) The diameter of the nucleus of the aluminum atom is 1.94 times smaller than that of the gold atom.

7) $395.54

8) 8.0 × 102 cm2

Word Problem

9) Possible answer: In a set of inequalities and/or equations, all the statements represent the situation.That means for a value to be a solution, it must make all the statements true since each one represents some aspect of the problem.

Homework Problems

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Today we rearranged equations to solve for a particular variable. Here’s an example!

The formula for investment at simple interest is l = Prt. Solve for r.

Solving for r means rewriting the equation so r is alone. To do this, divide both sides of the equation by Pt.

l = Pr tPt Pt

l = rPt

1) The formula for investment at a compound interest is A = P(1 + r)t. Solve for P.

2) The following formula describes speed where s = speed, d = distance, and t = time. Solve for t.Explain your thinking.

s = d t



3) The following formula describes discount where D = discount, R = the regular price, and S = the price after discount. Solve for R.

D = R – S

4) The formula for energy is E = mc2. Solve for m.

5) The formula for area of a circle is A = πr2. Solve for π.

6) A coordinate geometry formula is y = mx + b. Solve for x.

7) The formula for the volume of a pyramid is V = 1 Bh. Solve for B.3

8) The following formula describes work where W = work, f = force, and d = distance. Solve for d.

W = fd

Mixed Practice

9) Write an equation or inequality for the situation. Then solve.

7 is at least the sum of 10 and the product of 2 and a.

10) Graph the equation on graph paper: The sum of 4a and 5b is equal to 30.

11) Find the surface area. Round to the correct number of significant figures.

12) Jen is baking. She needs 1 cup of flour, but only has a teaspoon to measure. She knows there are4

16 tablespoons in 1 cup and 3 teaspoons in 1 tablespoon. How many tablespoons should she use?

Word Problem

13) The sale price, S, is equal to the difference between the list price, L, and the product of the list price and discount rate, r. Write a formula that solves for the list price.




Today your student rearranged formulas to solve for particular variables. Your student used what he or

she knows about the Order of Operations and solving equations with mathematical properties to help.

Rearranging formulas is helpful because it often makes finding the value of that variable easier.

For example, if you were finding the radius of multiple circles, it would be easier to arrange the formula

for the circumference of a circle C = 2πr to r = C . This also prepares your student for rewriting more

and2π

more complex equations.

Your student will use this to solve systems of equations in upcoming cycles.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about r ea rr a n g i n g e qua t i o n s :

1) Why might this equation need to be rewritten this way?

2) What properties do you need to use to rewrite the equation?

3) How does the Order of Operations help you rewrite the expression?

He r e a r e s o m e i d eas to w o r k w i t h r ea rr a ng i ng eq u at i o n s :


http s :/ / www. k hana c a d e m y . o r g/ m ath/a r i th m et i c / r ate s- a nd -r at i on s /u n i t_ c on v e rs i o n/ v / c o n v e r t i ng- f a r enhe i t - t o -c e l s i us

2) Review rearranging linear equations on:

http:/ / www . vi r t ua l ne r d. c o m /a l g eb r a - 1 / l i nea r- e qua t i o n s-s o l v e/ i s o l a t e - v a r i a b l e - f r o m - f o r m u l a.php

http:/ / www . vi r t ua l ne r d. c o m /a l g eb r a - 1 / l i nea r- e qua t i o n s-s o l v e/ i s o l a t e - v a r i a b l e s-i n - t e r m s- o f - v a ri a b l e s .p h p

Homework Answers

1) A = P(1 + r )t

2) t = d s

Possible explanation: I used inverse operations to isolate t on one side of the equation. I saw that I knew about the structure of this formula even though I don’t know any of the values of the variables. Because s equals the ratio of two values, I saw I could multiply both sides by the same terms to isolate t.

3) R = D + S 4) m =

5) π = A

E

c 2

y – b6) x =r 2 m

7) B = 3V 8) d = Wh f

Mixed Practice

9) 7 ≥ 10 + 2a; – 3 ≥ a2

10)

11) 13 cm2

12) She should use 12 teaspoons.

Word Problem

13) S = L – rLS = L(1 – r) S = L(1 – r )

Homework Problems

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Today we solved linear equations by justifying each step, showing the properties we used to keep the two sides of the equation equal. Here’s an example!

Solve the equation 2(x + 3) = –1 and justify each step.

2(x + 3) = –1 given

2x + 6 = –1 distributive property

2x + 6 + –6 = –1 + –6 addition property of equality

2x + 0 = –7 additive inverse property

2x = –7 additive identity property

1 – 1 (2x) = ( 7) multiplication property of equality 2 2

–1x = 7

2 multiplicative inverse property

x = –3 1

2 multiplicative identity property



Directions for questions 1–6: Solve the equation and justify each step.

1) –4x = 16 – 26 Explain your thinking.

2) 7 – 2 = 9x

3) 3x – 1 = 4 – 7x

4) –14 = 2 x – 63

5) 3 = 5x + 1

6) –4 – x = 1 (8 – x)2

Mixed Practice

7) Solve P = 2w + 2l for w.

8) What is the multiplicative inverse of 0.03?

9) Find the surface area of a cube with side lengths of 3.1 × 104 cm. Round to the correct number of significant figures.

10) Use the number line below to graph x ≥ 1 7 . Be sure to label each division on the number line.8

Word Problem

11) Explain why justifying each step in solving an equation is a useful process.




Today your student explained each step he or she took in solving linear equations. This practice of recording the properties used at each step is not only a good review of properties of operations, but it also helps your student reflect on a process he or she will be using over and over again this year. It also reinforces that we keep both sides of the equation equal at each step. This practice provides a solid basis for when your student is faced with much more complex equations with multiple variables and exponents, for example. As long as we preserve both sides as equal by applying the properties, the more complex equations are simple to solve as well.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about j u s t i f y i n g s o l ut i on s :

1) Explain how you solved for x in this equation.

2) What does this property mean, in your own words?

3) Is there a different way you could have solved this equation?

4) Are the two sides of this equation still equal after what you did? How do you know?

He r e a r e s o m e i d eas to w o r k w i t h j u s t i f y i ng t he s o l u t i o ns to e qua t i o n s :

1) With your student, take a look at a recipe or instructions for something that needs assembly. Discuss the benefits of a step-by-step approach. Analyze the recipe/instructions to evaluate how well each step is explained so that anyone following them would have the same result. Could the steps be in a different order? Could they be explained more simply?

2) Use Khan Academy to review solving equations: https://www.khanacademy.org/math/algebra/solving-linear-equations-and- inequalities/why-of-algebra/v/why-we-do-the-same--thing-to-both-sides--two-step- equations

3) Or, use Khan Academy to review solving equations with variables on both sides: https://www.khanacademy.org/math/algebra/solving-linear-equations-and- inequalities/why-of-algebra/v/why-we-do-the-same--thing-to-both-sides-multi-step- equations

4 4

2

2 2

2

2

Homework Answers

Note: All work shown for problems 1–6 is possible work.

1)–4x = 16 – 26 given

–4x = –10

– –

addition

1 (–4x) = (–10) 1 multiplication property of equality

1 01x = 4 multiplicative inverse property

x = 2 12

multiplicative identity property

Possible explanation: I used the operation of addition and the properties of multiplication to solve the

problem. Because I have an equation with variables, I know I need to isolate the variable on one

side

of the equation. First I added to simplify the right side of the equation. Then I used the multiplicative

property of equality to multiply each side by– 1 . The multiplicative inverse property tells me that I

4

canceled out the –4 on the left side, and the multiplicative identity property isolates the x.

2)7 – = 9 given

x

7 – 2 + –7 = 9 + –7 addition property of equalityx

0 – = 2 additive inverse propertyx

– 2 = 2 additive identity propertyx

– (–x)

x= (2)(–x) multiplication property of equality

(1)(2) = –2x multiplicative inverse property

2 = –2x multiplicative identity property

– – 1 (2) = (–2x) 1 multiplication property of equality

–1 = 1x multiplicative inverse property

–1 = x multiplicative identity property



3)3x – 1 = 4 – 7x given

3x – 1 + 7x = 4 – 7x + 7x addition property of equality

10x – 1 = 4 + 0 additive inverse property

10x – 1 = 4 additive identity property

10x – 1 + 1 = 4 + 1 addition property of equality

10x + 0 = 5 additive inverse property

10x = 5 additive identity property

1 1 (10x) = (5) multiplication property of equality10 10

11x = 2 multiplicative inverse property

1x = multiplicative identity property2

4)–14 =

–14 + 6 =

–8 =

–8 =

2 x – 6 given3

2 x – 6 + 6 addition property of equality3

2 x + 0 additive inverse property3

2 x additive identity property3

3 – 2 3 ( 8) = x multiplicative property of equality 2 3 2

– 2 4 = 1x multiplicative inverse property2


5) 3 x + 1 = 5 given

3 (x + 1) x + 1

= (5)(x + 1) multiplication property of equality

(1)(3) = (5)(x + 1) multiplicative inverse property

3 = 5(x + 1) multiplicative identity property

1

3 = 5x + 5 distributive property

3 + –5 = 5x + 5 + –5 addition property of equality

–2 = 5x + 0 additive inverse property

–2 = 5x additive identity property

1 – 1 ( 2) = (5x) multiplication property of equality 5 5

– 2 = 1x multiplicative inverse property5

– 2 = x multiplicative identity property5

6)–4 – x =

1 (8 – x) given2

–4 – x = 4 –

–4 – x + x = 4 –

–4 + 0 = 4 +

–4 = 4 +

–4 + –4 = 4 +

1 x distributive property2

1 x + x addition property of equality2

1 x additive inverse property2


1 x + –4 addition property of equality2

–8 =

–8 =

1 x + 0 additive inverse property2


(2)( –8) = x (2) multiplication property of equality 2

–16 = 1x multiplicative inverse property


Mixed Practice

7) w = P – 2 l 2



8) 10 03

9) 5.7 × 109 cm2

10) Possible graph:

Word Problem

11) Possible explanation: It’s useful to explain each step in solving an equation so you remember that you are doing the same thing to both sides at each step. This way you can see that the two sides of the equation stay equal all the way throughout.

Homework Problems

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justify, inequality

Today we practiced solving more complex linear equations. Here is an example!

Solve for x.

9(x – 3) = 5 – ax

9x – 27 = 5 – ax

9x – 27 + 27 = 5 – ax + 27

9x + 0 = 32 – ax

9x = 32 – ax

9x + ax = 32 – ax + ax

9x + ax = 32 + 0

9x + ax = 32

We used the properties to rearrange the equation so the variable weneed to solve for is on one side ofthe equation.

At this point, we can use the distributive property to factor out x from both addends on the left. Then we continue to solve for x.

x(9 + a) = 3 2 (9 + a) (9 + a)

x = 3 2 9 + a



Directions for questions 1–7: Solve for x.

1) 7x + 4 + ax = 25 2) 3x + 8 = 9 – x + 1

3) ax + 7 = 3 – bx + ax 4) 1 (4 – 2x) = 3x – 142

5) 5 – ax = x + 2 6) 1 – b = 11x

7) 16 = 2 – x a

Explain your thinking.

Mixed Practice

8) Solve the equation– 1 x = 15 and justify each step.

3

9) Train A travels 210 miles in x hours. Its speed is half of Train B, whose speed is 88 miles per hour.Solve for x.

10) This year, Mr. McGill’s school has 835 students. Last year, it had 913 students. By what percent did the enrollment decrease?

11) A model drawing of a painting has a length of 5.25 inches and a width of 8 inches. The actual drawing has a length of 3 feet. What is the width of the actual painting?

Word Problem

12) Write an equation for the math statement and solve for x.The product of 5 and the sum of x and 1 is equal to x.


Today your student practiced solving more complex linear equations for a single variable. Your student used what he or she knows about the properties to solve equations. Your student also used his or her knowledge of working with variables and exponents to solve more complex equations. This will prepare him or her to find solutions to inequalities in the next lesson.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e s e q ue s t i o n s about s o l v i ng e q u at i ons f or a s i n g l e v a r i a b l e

1) Explain how you solved for x in this situation. What properties helped you?

2) How did you keep the equation balanced while you were solving for x?

3) Will there be variables in your solution? How do you know?

4) How can you check your work?

5) Is there a different way you could have solved this equation? Show me.

He r e a r e s o m e i d eas to w o r k w i t h j u s t i f y i ng t he s o l u t i o ns to e qua t i o n s :

1) Use Khan Academy to review solving equations with the distributive property: https://www.khanacademy.org/math/algebra/solving-linear-equations-and- inequalities/complicated_equations/v/solving-equations-with-the-distributive-property

2) Use Khan Academy to review solving equations with variables on both sides: https://www.khanacademy.org/math/algebra/solving-linear-equations-and- inequalities/basic-equation-practice/v/solving-equations-2

3 1 x 3



Homework Answers

1) x = 21 7 + a

2) x = 12

–3) x = 4

b

4) x = 4

5) x = 3 1 + a

6) x =– b 10

7) x = –16a + 2

Possible explanation: I used the properties of multiplication and addition to solve for x. Because I had to isolate x on the right side of the equation, I knew that I had to first use the multiplication property of equality to multiply both sides by a. Then I used the addition property of equality to add –

2 to each side. This left me with 16a – 2 = –x. The multiplicative identity property helped me know that multiplying both sides by –1 would get me x = –16a + 2.

Mixed Practice

8)

– 1 x = 15 given3

– –

– = 15 multiplication property of equality

1 3 1

(1)(x) = –45 multiplication inverse property

x = –45 multiplication identity property

9) x = 4.77 hours

10) The enrollment decreased by 8.5%.

11) The width of the actual painting is 4.6 feet long.

Word Problem

12) 5(x + 1) = x x = –1.25

5 5

Homework Problems

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justify, inequality

Today we practiced solving inequalities in one variable. Solving an inequality simply means finding the solution set for the variable that makes the inequality true. That is, we isolate the variable on one side of the inequality. Here’s an example!

4(x – 5) ≥ –x

4x – 20 ≥ –x

4x – 20 – 4x ≥ –x – 4x

0 – 20 ≥ –5x

–20 ≥ –5x

We use the properties we know to simplifythe expressions on each side and combine terms to get x on one side.

– – 1 (–20) ≥ (–5x) 1

4 ≤ 1x

4 ≤ x

Notice that when we multiply or divide both sides of an inequality by a negative number, we have to–switch the sign. Why? Think about how 3 < 8 is a true statement, but multiplying both sides by

it untrue.1 makes



1) Solve for x and graph the solution. Explain your thinking.

2 – x < 7(1 + x)

2) Solve for x and graph the solution.

5 – 3x ≥ 1 – x

3) Jackson needs to buy 3 chew toys for her 3 dogs. They have to be exactly the same toys because her dogs are very particular. Her budget is $28. Sales tax in Jackson’s city is 5%. Write an inequality for this situation. Solve the inequality to find the highest-priced chew toy she can buy.


–2x ≥ 7 – x


4(x + 3) > 2 – x


1 x ≥ 6 – x2

Mixed Practice

7) Solve for x.

6 – ax = 4 + bx

8) If a car travels 199.5 miles in 3.5 hours, what is its average speed?

9) Find the circumference of a circle with a radius of 9.0 inches. Round to the correct number of significant figures.

10) The formula for a cone is V = 1 πr2h. Solve for h.3

Word Problem

11) Explain how solving linear inequalities is similar to and different from solving linear equations.




Today your student solved inequalities in one variable, like 11 – 2x > 4x. Solving inequalities is very similar to solving equations, except you must remember an important rule about multiplying or dividing with negative numbers—the inequality sign is reserved when each side of the inequality is multiplied or divided by the same negative number. Practice solving inequalities is helping prepare your student for more complex algebra later on this year. The work your student has done in this cycle should reinforce that complex equations and inequalities can be simplified and represented by a simple graph or solution set.

Y our s t u de n t s ho u l d be a b l e to an s w er t h e f o l l o w i ng q ue s t i o ns a b out s o l v i n g i ne q ua l i t i e s :

1) Explain how you solved this inequality.

2) Why did you switch the direction of the sign?

3) Does your answer make sense? How do you know?

4) How would your solution be different if this were an equation and not an inequality?

T o he l p y our s t ud e nt p r a c t ic e s o l v i n g i ne q u a li t i es y ou m i ght li k e to:

1) Represent everyday situations with inequalities. For example, perhaps you’ve budgeted that your monthly food bill should not go above a certain number. Write an inequality with your student that represents this situation and together consider which costs are variable each month and which are fairly stable.

2) Watch a Khan Academy video about solving inequalities. http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / i ne q u a li t i e s / v / i ne q ua l i t i e s- u s i ng- m u l t i p li c a t i o n - a n d - d i vi s i on

3) Watch another Khan Academy video about solving inequalities. http s :/ / www. k hana c a d e m y . o r g/ m ath/a l g eb r a / l i nea r _ i ne qu a li t i e s / i ne q u a li t i e s / v / m u l t i-s tep- i n e qu a li t i es

Homework Answers

–1) x > 5

8

Possible explanation: I used the properties of addition and multiplication to isolate the variable on one side of the inequality and solve. I knew that I could look at the structure of the problem and use the properties of multiplication and addition to complete each step. I didn’t have to reverse the inequality because I did not multiply or divide by a negative number.

2) x ≤ 2

3) 3x + 0.05(3x) ≤ 28, x ≤ 8.89, so she has to choose a toy that is less than or equal to $8.89.

4) x ≤ –7

5) x > –2

6) x ≥ 4

Mixed Practice

7) x = 2 8) Its average speed is 57 miles per hour.b + a

9) 57 in. 10) h = 3V πr 2

Word Problem

11) Possible explanation: We use the same properties to solve for x no matter whether we’re solving an equation or an inequality. But with an inequality, we aren’t keeping both sides equal; we are keeping the same unequal relationship. I can use the structures of equations and inequalities to help me understand their differences. In an inequality, one side, for example, is greater than the other. So if we multiply both sides by a negative number, we have to change the sign because that multiplication switched which side was greater.

Homework Problems

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Today you and your team started planning a menu for a school lunch option. You learned about the daily recommendations for nutrition, and the recommendations for nutrition set out by the National School Lunch Program. These recommendations create some constraints for your meals. They cannot go over a certain amount of calories, have too much fat or salt, or need to contain a certain amount of protein.



To figure out how much of which ingredients you could use, you represented the constraints with inequalities and/or equalities and plugged in values to see whether you could have as much of an ingredient as you wanted. Sometimes you could add more of that ingredient; sometimes you had to take some away.

Directions for questions 1–6: Use the charts above and below to solve the problems.

1) Allanah wants a lunch that contains up to 600 calories because she wants dessert after dinner. She is ordering a bacon cheeseburger, two orders of apple slices, and a soda. Write an inequality and/orequality that represents this situation. Explain whether she will meet her lunch goal.

2) Balik wants a bacon cheeseburger for lunch. How much sodium should he limit himself to for the rest of the day?

3) Selena only wants to eat 1 of her maximum daily calorie limit at lunch. She orders a Chicken3

Doodlewich, an order of French fries, and a Chocolate Shake-a-Doodle.

a. Write an inequality and/or equation that represents this constraint.

b. Does her lunch keep the inequality and/or equation true?

4) Does a lunch of chicken doodles, 1 order of French fries, and a Vanilla Shake-a-Doodle satisfy the limits for daily intake of saturated fats?

5) If Billie eats a MacDoodle Fishwich, 1 order of apple slices, and a Chocolate Shake-a-Doodle, is the amount of protein he has eaten greater than or less than the daily recommended amount? Write aninequality and/or equation to represent this problem.

6) It is recommended that people consume 0 grams of transfat each day. Write an inequality and/or equation that represents this constraint. Which menu items should be avoided at MacDoodle’s to make this true?

Mixed Practice

7) Hippopotamuses can sprint over short distances on land at 30 miles per hour. There are 5,280 feet in1 mile. If a hippo runs 97 feet to scare away a predator, how many minutes did it take the hippo to reach its stopping point?

8) A pyramid has a volume of 1,365 cm3.

The formula for finding the volume of a pyramid is V = 1 • l • w • h. What is the value of h?3

9) Solve the equation.

5x + 7 = 1 3 + 4x




–8 + 4x ≤ 3 – 2x

Word Problem

11) How can you represent constraints for nutrition when you are given a range of values, such as for calorie intake?


Today your student applied his or her knowledge of representing constraints with inequalities and equations to a real-world context. Your student started a three-day project in which he or she will explore what is involved in planning a school lunch menu. Throughout this three-day project, your student will look at nutritional data and calculating the nutrition of food items, costs, and other dietary considerations when picking food options for a lunch.

These periodic performance task cycles this year are intended to introduce your student to real-life examples where math is used to model what’s happening and to solve problems. The tasks in these cycles involve a lot of reading and making sense of a real-world situation. To help your student, you might like to explore more about nutritional information and dietary guidelines at the library or on the web. You also might ask your student to explain how to read a nutrition label from packages of food aroundthe home.



Homework Answers

1) cbacon + 2capple + csoda ≤ 600; Allanah will not meet her lunch goal.Possible explanation: I plugged in the numbers for the calories in the food she eats to see whetherthe inequality would be true. I translated the problem into math to figure out whether Allanah was meeting her goals. My work shows that 440 + 2(15) + 140 = 610. This is not less than or equal to 600. So Allanah has to adjust what she eats to meet her goal.

2) He should limit himself to 390 mg of sodium for the rest of the day.

3) a. cchick + cfries + cshake ≤1 • 18003

b. No. She will eat 1,150 calories with that lunch. That is greater than 1 of 1800, or 600.3

4) Yes. That lunch has 130.5 calories from saturated fat, which is less than 10% of 1600, or 160.

5) pfish + pshake ≥ 1 oz.; No, Billie only eats .945 oz. of protein in his meal. He needs to eat more protein throughout the day.

6) t = Trans Fat, t < 0; You should avoid eating the cheeseburger, bacon cheeseburger, and the chocolate and vanilla Shake-a-Doodles, since they each have more than 0 grams of Trans Fat.

Mixed Practice

7) It takes about 0.04 minutes for a hippopotamus to run 97 ft.

8) h = 13 cm

9) x = 1 7 or 1 5 12 12

10) x = 1 12

Word Problem

11) Answers will vary.

Homework Problems

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Today you and your team continued to explore what is involved in planning a school lunch. You started taking costs into consideration today as you planned your meals. The school district gets some foods subsidized through grants and other funds to help keep the cost of a lunch low, at $2.40. You were also introduced to the idea that a school lunch may consist of more than just a main meal item, but a drink or a side dish may come with it as part of the total price. This affects your nutritional constraints and yourcost constraints.



Directions for questions 1–4: Use the charts above and below, in addition to previous charts, to solve the problems.

1) Jacee has $10 to spend on lunch. She wants a hoagie with double meat and a soda with her lunch, but she wants to consume at most 700 calories.

a. Write inequalities and/or equations that represent the constraints for cost and calories.

b. Which hoagie options from the menu will make your inequalities and/or equations true? Explain your thinking.

2) Lorne orders a roast chicken hoagie on a whole wheat roll with Swiss cheese. He is watching his sodium intake, so he wants to make sure he eats less than 1,035 mg of sodium at lunch.

a. Write an inequality and/or equation that represents the constraint for sodium and tell whetherLorne’s hoagie keeps the inequality true.

b. Lorne has $2 extra that he wants to spend on a drink, snack, or dessert to go with his hoagie.Which option will keep him within the constraint for sodium?

3) Mondoe orders a roast beef hoagie on a parmesan roll with double meat and double Swiss cheese, and adds a chocolate chip cookie to his order. He has $13 in his pocket to pay for lunch. He also wonders whether his lunch is meeting the recommendations for saturated fats in his diet, based on the NSLP recommendations.

a. Write inequalities and/or equations that represent the constraints for cost and saturated fat.

b. Does Mondoe’s lunch make the inequalities true?

4) Deepika has $8 for lunch and wants to get a hoagie and either a drink, chips, or dessert. Which menu options will get her the most for her money in terms of protein and cost?

Mixed Practice

5) In a right triangle, side a is 12 cm and the hypotenuse (c) is 20 cm. Find b2.

6) Solve this system of equations.

2x + 5y = 577x – y = 33

7) a is less than or equal to 270 decreased by 5 to the power of x. Write an equation for the situation.

8) Solve for x.

x(4 – b) = 2x + 0.5



Word Problem

9) Think about some of the menu items you have looked at. How does taking the cost of these items into account affect other situations, such as the calories, protein, or sodium in the meal? Why do you think it is important to consider costs when planning a school lunch menu?


Today your student continued the performance task cycle on planning a school lunch menu. Your student worked with his or her team to learn more about considerations he or she needs to take into accountwhen planning a menu.



Homework Answers

1) a. Calories: cbread + 2cmeat + csoda ≤ 700Cost: mhoagie + mmeat + msoda ≤ 10

b. A turkey hoagie with double meat on an Italian roll with a soda meets both of the constraints. Iplugged numbers from the chart into my inequalities to see what would make them true. I builtmath models to help me figure out what Jacee could eat. For calories, 200 + 2(120) + 260 = 700.This made the inequality for calories true. For cost, 5.50 + 2.00 + 1.75 = 9.25, this is less than 10.That made the inequality for cost true.

2) a. Sodium: sbread + schicken + sSwiss < 1,035Yes, Lorne’s hoagie keeps the inequality true. 280 + 660 + 30 = 970, this is less than 1,035.

b. A soda will keep him under 1,035 mg of sodium.

3) a. Cost: mhoagie + mmeat + mcheese + mcookie ≤ 13Sat. fat: fbread + 2fmeat + 2fcheese + fcookie < 0.10 • total lunch calories

b. The inequality for price is true, but not for saturated fat. Mondoe only spends $10.50 on lunch.But he eats 135 calories from saturated fat. That is more than 10% of the total calories ofhis lunch.


Mixed Practice

5) b = 16

6) (6, 9)

7) a ≤ 270 – 5x

0 . 5 8) x =2 – b

Word Problem


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