Unit 2 Cycle 1 - Adding and Subtracting Rational … | P a g e Unit 2 Cycle 1 - Adding and Subtracting Rational Numbers Lesson 2.1.1 - Definition of Rational Numbers Vocabulary integers

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Unit 2 Cycle 1 - Adding and Subtracting Rational Numbers

Lesson 2.1.1 - Definition of Rational Numbers

Vocabulary

integers rational numbers natural numbers

whole numbers terminating decimal repeating decimal

Active Instruction

ta(1) Rational numbers are numbers that can be written in the form of a fraction.

ta(2) What are four rational numbers that fall between

and 1?

th(1) Write four numbers that fall between

and

.

Team Mastery

(4) Write

three different ways.

For the Guide on the Side

Today your student defined rational numbers, then used the operations and number lines to explore them. A rational number is a number that can be written in fractional form, so , –3.5, and 19,000 are all rational

numbers. Rational numbers are whole numbers, repeating and terminating decimals, fractions, and integers.

Once your student better understood rational numbers and how they compare to integers, whole numbers, and natural numbers, he or she used number lines and number sense to write rational numbers whose value fall between given rational numbers or wrote given rational numbers in new ways. For example, can be written as ,

0.500, 50%, or 1 – among many other ways.

Now that your student understands rational numbers better, it will be easier for him or her to combine those different types of rational numbers in the upcoming lessons and units. We will be learning how to add, subtract, multiply, and divide all rational numbers, include negative rational numbers.

Your student should be able to answer the following questions about rational numbers: 1) Why is this not a rational number? 2) How else could you write this number? How do you know it has the same value? 3) 3.4 and 3.5 are two numbers between 3 and 4; can you name two numbers between 3.4 and 3.5?

Here are some ideas to work on rational numbers: 1) Video: How do different categories of numbers compare to each other?

http://www.virtualnerd.com/algebra-2equations-inequalities/real-numbers/number-types/numbercategory-comparison

2) Find two items that are close to each other in price at the grocery store, then think of prices or other numbers that would come between those two prices.

3) Use your student’s lucky number/age/number of students in his or her class and have your student rewrite that number as many ways as possible.

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Lesson Quick Look

Team Name Team Complete?

Team Did Not Agree On

Questions…

Write the vocabulary introduced in this lesson:

Today we defined and explored rational numbers. We learned how rational numbers compare to natural numbers,

whole numbers, and integers.

For example, 14 is a natural number, a whole

number, an integer, and a rational number. 0 is a

whole number, an integer, and a rational number.

While –7 is an integer and a rational number, 0.25 is

only a rational number.

Finally, we explored rational numbers using a number

line. We know that there are many rational numbers

between –1.2 and –1.3.

Lesson 2.1.1 - Homework

Directions for questions 1 and 2: Write four numbers that fall between the given numbers. Explain your thinking.

1) – and –

2) 0.123 and 0.124

Directions for questions 3 and 4: Solve. Classify the solution as: natural number, whole number, integer, rational number. (Use all that apply.)

3)

•

4) 25 ÷ 0.8

Directions for questions 5 and 6: Write each number

three different ways.

5) –

6) 0.60

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Lesson 2.1.2 - Adding Opposites

Vocabulary

opposite zero pairs

Active Instruction

ta(1) What does it look like to add +1 + –1?

ta(2) Raphael ran 60 feet toward first base. Then he heard the umpire yell "foul ball!" so he ran back to home

plate. What is Raphael's distance from home plate now?

th(1) Sayde borrowed $20 from her brother. When she got her allowance, she paid him back $20. How much does Sayde owe her brother now? Team Mastery

(5) Cedric lives 6 blocks from his school. He walked halfway there then realized he left his permission slip at home. So he went back home to get it. How far is Cedric from his home now? For the Guide on the Side

Today your student worked with rational numbers and their opposites. Opposites are numbers that are the same

distance from zero on a number line but in the opposite direction. Opposites are also numbers, that when added,

give a sum of 0. Your student translated math stories into addition sentences involving opposites. Then he or she

solved the addition sentence to answer the math question.

This lesson helped your student better understand what happens when positive and negative numbers are

combined. As we continue to work in this cycle, we will learn to combine more complicated numbers as well as

more than two numbers.

Your student should be able to answer the following questions about adding opposite numbers for a sum of zero.

1) What is the opposite of this number?

2) How can you write this situation mathematically?

3) Describe what is happening in this situation.

Here are some ideas to work on adding opposite numbers for a sum of zero:

1) Video: Inverse Property of Addition: https://www.khanacademy.org/math/arithmetic/order-ofoperations/arithmetic_properties/v/inverse-property-of-addition

2) Website: Properties Lesson 6 – The Additive Inverse Property: http://www.coolmath.com/prealgebra/06-properties/07-properties-additive-inverse-01.htm

3) Video: Write equivalent expressions using Additive Inverse Property:

http://learnzillion.com/lessons/650-write-equivalent-expressions-using-the-additive-inverseproperty

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Lesson Quick Look



Questions…


Today we worked with rational numbers and their opposites. Opposites are numbers that are located the

same distance from zero on the number line, but in opposing directions. We can use zero pairs to understand

adding opposite numbers. +1 and –1 combine for a sum of 0. So 25 positives and 25 negatives would also

combine for a sum of 0.

Then we used our knowledge of opposites to help us describe math stories. Here’s an example!

Kendrick drives 35 miles to work in the morning. He takes the same route to get home in the evening. How

many miles is Kendrick from his home after his evening drive?

35 + (–35) = 0

Kendrick drives 35 miles to work, then 35 miles in the opposite direction. Now he is 0 miles from home

after his evening drive.


1) Jared and his Dad were practicing throwing a baseball with twenty feet between them. Dad throws the ball to Jared, and Jared throws it back. What’s the distance now between Dad and the baseball? 2) Juanita runs down a ten-foot flight of stairs from the second floor. Then she runs back up because she forgot something. How many feet is Juanita from the second floor when she returns to the top of the stairs?

3) Kai passed the “start” square on a board game

and earned 100 game dollars. In the same turn, he landed

on a penalty square which costs 100 game dollars. How

much money did he earn in all during his turn?

4) An atom has 6 electrons, which each have a

negative charge. It also has 6 protons, which each have a

positive charge. What is the overall charge of the atom?

5) Jamie’s dog Rex ran fifteen meters to chase the

ball Jamie threw. Then Rex returned the ball to Jamie. How

far is Rex from Jamie now?

6) Scientists sent a robot ship down 12,500 feet to the resting place of the Titanic. Then the robot ship returned to the surface. How far was the robot ship from the surface when it returned? Mixed Review 7) Write 8/15 three different ways.

8) Write ten numbers that fall between 0.325

and 0.326

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Lesson 2.1.3 - Adding Rational Numbers 1

Active Instruction

ta(1) Use a number line to find the sum of 16 + (–20).

ta(2) Draw a picture to find the sum of (–2) + 13.

ta(3) What's the sum of (–1) + (–7)?

th(1) Use the number line to find the sum: (–23) + (–32)

Team Mastery

(4) Draw a picture to find the sum: 8 + (–22)


Today your student added integers using manipulatives and drawings like number lines, algebra tiles, and charged fields. We worked with both positive and negative numbers and added numbers with like and unlike signs. If we were to find the sum of 32 + (

–19), we could draw a picture or use a number line and get the same sum, 13.

For example, with charged fields and algebra tiles eliminate the zero pairs and count the remaining charges as our solution.

On a number line, we start at zero then move in the positive or negative

direction depending on the addends.

In the next lesson, your student will use algorithms to add integers. He or she will use their knowledge from this lesson to explain their results.

Your students should be able to answer the following questions about adding integers using manipulatives and drawings:

1) How can you show this problem on a number line?

2) Why do you eliminate “zero pairs”?

3) How did you find the sum for this problem? Here are some ideas to work on adding integers with manipulatives and drawings:

1) Video: Adding negative numbers:

https://www.khanacademy.org/math/arithmetic/absolutevalue/adding_subtracting_negatives/v/adding-negative-numbers

2) Video: Add integers using a number line:

http://learnzillion.com/lessons/1698-add-integers-using-a-number-line

3) Video: Add positive and negative integers using a number line:

http://learnzillion.com/lessons/1699-add-positive-and-negative-integers-using-a-number-line

4) Website: Algebra Tiles:

http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html

5) Website: Algebra Tiles: http://www.x-power.com/Flash/Tools/AlgebraTiles.html

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Lesson Quick Look



Questions…


Today we added integers using manipulatives and pictures. We worked with both positive and negative numbers.

We use number lines to show the direction of each addend as well as the final answer. For example, if we add 14 + (–21), our number line would look like this:

We can see on the number line that if we first add 14, then add –21, the sum is –7.

We also used algebra tiles or charged fields to help us eliminate the zero pairs and find the answer.

When we eliminate all the zero pairs in 14 + (–21), we can see that there are 7 negative charges left, so the sum

is –7.


1) (-6) + (-4)

2) (-16) + (-5)

3) (17) + (+12)

Draw a picture to find the sum.

4) (+13) + (-5)

5) 12 + (-21)

6) (-7) + 13

Mixed Review

Find the difference.

7) 95.5 – 81.68

8)

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Lesson 2.1.4 - Adding Rational Numbers 2

Vocabulary

absolute value

Active Instruction

ta(1) Find the sum:

(

)

th(1) Write an addition sentence to answer the question. Explain your thinking. A U.S. Navy submarine was submerged 800 feet below sea level. It ascended 375 feet. Then it ascended 250 feet more. What is the submarine's depth now?

Team Mastery

(5) Find the sum and show your work. Explain your thinking.

(

)


Today your student learned the algorithm to help them add rational numbers with like and unlike signs. This built on your student’s knowledge of combining rational numbers. Now instead of using number lines and algebra tiles, your student can use simple steps to help them find the sum. This is helpful when the numbers combined are more complex like fractions and decimals.

When adding rational numbers with like signs: 1) Find the absolute value of each rational number. 2) Find the sum. 3) Use the sign of the addends for the sum.

When adding rational numbers with unlike signs: 1) Find the absolute value of each rational number. 2) Subtract the addend with the lesser absolute value from the addend with the greater absolute value. 3) Use the sign of the addend with the greater absolute value for the sum.

In the next cycle, your student will continue to work with rational numbers by subtracting them.

Your student should be able to answer the following questions about adding rational numbers: 1) How do you know what the sign of the sum should be? 2) What’s the difference between adding like signs and adding unlike signs? 3) How did absolute value help you find the sum? 4) Which numbers will you add first for this problem?

Here are some ideas to work on adding rational numbers: 1) Video: Adding integers with different signs:

https://www.khanacademy.org/math/arithmetic/absolutevalue/adding_subtracting_negatives/v/adding-integers-with-different-signs

2) Video: Pre-algebra Lesson 11: Adding Integers: http://www.youtube.com/watch?v=LGe2SOAA9F0 3) List the income and expenditures for a week for a high school student working at a local business. Did the student

spend money or save money this week? 4) Math Goodies: Adding Integers: http://www.mathgoodies.com/lessons/vol5/addition.html 5) Math Play: Adding Integers Game:

http://www.math-play.com/adding-integers-game/adding-integers-game.html 6) Kuta Software: Free Adding Rational Numbers Worksheet:

http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Adding+Subtracting%20Rational%20 Numbers.pdf

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Lesson Quick Look



Questions…


Today we added rational numbers with like and unlike signs. Here’s how we do it!

When adding rational numbers with like signs:

1) Find the absolute value of each number. 2) Add the numbers. 3) Use the sign of the addends for the sum.

When adding rational numbers with unlike signs:

1) Find the absolute value of each rational number. 2) Subtract the addend with the lesser absolute value from the addend with the greater absolute value. 3) Use the sign of the addend with the greater absolute value for the sum.

**When working with three or more addends (with unlike signs) remember to combine the like terms before subtracting.

Knowing how to add rational numbers with like and unlike signs can help us with math in the real world as well. For example, it can help me find the balance in my savings account at the bank!

Transaction

Number Date Description Withdrawal Deposit Balance

1 1/15 Payroll $423.00 $423.00

2 1/16 Donation $142.30 $280.70

3 1/17 Shopping trip $296.85 –$16.15

4 1/18 Tips $135.76 $119.61


1) 24.65 + (–32.987)

2)

(

)

3) (-30.413) + (-651.021)

4)

5) Because of the sudden melting of heavy snows, the Fraser River flooded, rising 2.3 feet on Monday, 1.5 feet on Wednesday and then fell 0.75 feet on Friday. What was the overall change in height of the river?

6) A parrotfish swims at a depth of 6 feet over a coral reef in the Florida Keys. The parrotfish dives down two feet to take a bite of coral. A shark appears, and the parrotfish swims up 1.5 feet to hide in a crevice in the reef. At what depth is the parrotfish now?

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Lesson 2.2.1 - Subtracting Integers

Vocabulary

addend additive inverse

Active Instruction

ta(1) Find the difference:

ta(2) For any numbers p and q, p – q is the answer to q + = p.

th(1) Find the difference. Show your work:

Team Mastery

(4) Find the difference. Show your work:


Today your student subtracted integers using manipulatives like algebra tiles and drawings like number lines and charged fields. These are the same tools we used when we added integers!

Today we worked with both positive and negative numbers and subtracted numbers with like and unlike signs. If we were to subtract 12 – (–9) using a number line, we would show 12 first.

Then we would show subtracting –9. Subtracting means we go to the left on the number line. BUT the negative sign tells us to do the opposite and move to the right instead.

We can see the difference is 21!

In the next lesson, your student will use algorithms to subtract integers. He or she will use their knowledge from this lesson to explain their results.

Your student should be able to answer the following questions about subtracting integers with manipulatives and drawings: 1) How can you show this problem on a number line? 2) What do you do if you don’t have enough tiles to take away? 3) 3) How do you find the difference for this problem?

Here are some ideas to work on subtracting integers with manipulatives and drawings: 1) Video: How do you Subtract Integers using a Number Line?: http://virtualnerd.com/middlemath/integers-coordinate-

plane/subtract-integers/subtract-integers-number-line-example

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2) Video: Understand subtraction as adding the additive inverse: http://learnzillion.com/lessons/675understand-subtraction-as-adding-the-additive-inverse

Lesson Quick Look



Questions…


Today we subtracted integers using manipulatives and pictures. We worked with both positive and negative numbers.

We can use number lines to find the difference. Remember, the minus sign means move to the left on the number line. However, if the number behind the minus sign is negative we do the opposite and move right. For example, if we subtract 4 – (

–

4), our number line would look like this:

We can see on the number line that we start at zero and move to 4. The second arrow starts at 4 and moves 4 spaces to the right. Normally to subtract we go to the left, but we are subtracting a negative so we do the opposite and go right, or in the positive direction. The difference is 8.

We can also use algebra tiles or charged fields to find the difference. When we did not have enough to subtract the number behind the minus sign, we added zero pairs, and then eliminated the number behind the minus sign to find the difference. If

we subtract 4 – (–4), we start with 4 positive tiles. Then we add 4 zero pairs so we have enough negative tiles to subtract 4 of

them. Then we eliminate 4 negative tiles to solve. We can see that the difference is still 8, because we are left with 8 positive tiles.


1) 2 – (–12)

2) (–7) – (–6)

3) 0 – 9

Draw a picture to find the difference.

4) (–5) – (–9)

5) (-13) – 8

Mixed Review

6) Order from least to greatest:

3.012 3.011 3.001 3.100 3.021

7) Write 3 ¼ three different ways.

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Lesson 2.2.2 - Use Properties to Subtract Rational Numbers

Active Instruction

ta(1) Use a number line and algebra tiles to subtract

ta(2) Use a number line to subtract

th(1) Draw a picture to find the difference. Explain your thinking.

Team Mastery

(8) Use a number line to show the difference. Explain your thinking.


Today your student subtracted rational numbers by solving for a missing addend and using the additive inverse of the problem. The additive inverse of a number is the same number, but of the opposite sign.

We discussed the additive inverse by thinking of any number p and q, and substituting p – q as a missing addend in the expression q + (p – q) = p. If we know that p – q = p + (–q), then we know that q + (p + (–q)) = p, too.

q + (p – q) = p

q + (p – q) = p

p + 0 = p p = p

q + (p + (–q)) = p

q + (p + (–q)) = p

p + 0 = p

p = p

Knowing that p – q = p + (–q) gives your student the tools he or she needs to solve subtraction problems by thinking of them as addition. We can use the additive inverse to subtract rational

numbers.

What your student learned today creates a foundation for the next lesson, where students will use algorithms to solve subtraction problems. It is easy to use the shortcut “add the opposite,” but the goal of today’s lesson was to help students understand why adding the opposite works.

Your student should be able to answer the following questions about subtracting rational numbers by finding the missing addend and using the additive inverse:

1) How can you show this problem on a number line? 2) What do you do if you don’t have enough tiles to take away? 3) How do

you find the difference for this problem?

Here are some ideas to work on subtracting rational numbers by finding the missing addend and using the additive

inverse:

1) Video: Understand subtraction as adding the additive inverse. http://learnzillion.com/lessons/675-understand-subtraction-as-adding-the-additive-inverse

2) Video: What is the opposite, or additive inverse, of a number? http://www.virtualnerd.com/tutorials/?id=PreAlg_01_02_0004

3) Video: How do you rewrite subtraction as addition? http://www.virtualnerd.com/pre-algebra/algebra-tools/add-subtract-integers/subtractintegers/subtraction-rewritten-as-addition

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Lesson Quick Look



Questions…


Today we discussed subtracting rational numbers by rewriting subtraction problems to solve for the missing addend. That step prepared us to use the additive inverse of a subtraction expression to solve problems. Remember that the additive inverse of a number is the same number, but of the opposite sign. The additive inverse property shows that the sum of a number and its additive inverse is equal to 0.

We can understand the additive inverse by thinking of any number p and q, and substituting p – q as a missing addend in the expression q + (p – q) = p. If we know that p – q = p + (–q), then we know that q + (p + (–q)) = p, too.

q + (p – q) = p

q + (p – q) = p

p + 0 = p p = p

q + (p + (–q)) = p

q + (p + (–q)) = p

p + 0 = p

p = p

Knowing that p – q = p + (–q) gives us the tools we need to solve subtraction problems by thinking of them as an easier operation, addition. We can use the additive inverse to subtract rational numbers.


Directions for questions 1–6: Find the difference. Show your work.

1) 15.3 – (–11) Explain your thinking.

2) 24.5 – 16

3) –31 – (–8.6)

4) –16.4 – 14.6

5) 13 – 19.2

6) –6.8 – (–6.4)

Mixed Review

7) Use a number line to find the sum.

–4 + –8

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Lesson 2.2.3 - Subtracting Rational Numbers

Active Instruction

ta(1) Find the difference: 10.8 – 25.17

th(1) Cherrie drove

miles per hour (mph) above the speed limit. Devon drove

mph below the speed limit.

What is the difference between their driving speeds?

Team Mastery

(5) Find the difference. Show your work:


Today your student learned the algorithm to help him or her subtract rational numbers. This builds on your student’s knowledge of adding rational numbers and subtracting rational numbers using manipulatives. Now instead of using number lines and algebra tiles, your student can use simple steps to help them find the difference. This is especially helpful when the numbers are more complex like fractions, decimals, and greater numbers like 25,944.

When subtracting rational numbers, the rule is: Add the opposite. 1) Change the operation from subtraction to addition. 2) Change the sign of the second number to the opposite. 3) Solve.

In the next lesson, your student will continue working with rational numbers by adding and subtracting them. They will use the algorithms for adding and subtracting numbers as well as the Properties of Addition and the Order of Operations to help them. Your student should be able to answer the following questions about subtracting rational numbers:

1) What is the opposite of this number? 2) How could you rewrite this as an addition problem? 3) How is subtracting rational numbers similar to adding rational numbers?

Here are some ideas to work on subtracting rational numbers: 1) Video: How do you rewrite subtraction as addition?:

http://virtualnerd.com/middle-math/integers-coordinate-plane/subtract-integers/subtractionrewritten-as-addition

2) Video: Why Subtracting a Negative is Equivalent to Adding a Positive: https://www.khanacademy.org/math/arithmetic/absolutevalue/adding_subtracting_negatives/v/why-

subtracting-a-negative-equivalent-to-adding-a-positive 3) Video: How do you subtract a positive number from a negative number?:

http://virtualnerd.com/middle-math/integers-coordinate-plane/subtract-integers/subtract-positivefrom-negative-number

4) Video: How do you subtract a negative number from a positive number?: http://virtualnerd.com/middle-math/integers-coordinate-plane/subtract-integers/subtract-negativefrom-positive-number

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Lesson Quick Look



Questions…


Today we subtracted rational numbers using rules to help us. Here are those rules! When subtracting rational numbers, the rule is: Add the opposite. Step 1: Change the operation from subtraction to addition. Step 2: Change the sign of the second number to the opposite. Step 3: Solve. How can the rules help us solve this problem? What is the difference of –18 – (–6)? Step 1: –18 + (–6) Step 2: –18 + (+6) Step 3: –12 Subtracting rational numbers can be tricky, but when we change them to addition problems, we make problems that are easier and more familiar to us!


Directions for questions 1–6: Find the difference. Show your work.

1) Write a subtraction sentence to answer the question. Explain your thinking.

Max has $15 but wants to buy a video game for $49. What is the difference between what Max has and what he wants to spend?

2) –26.17 – (+18.52)

3) –18 – (–21)

4)

5) Write a subtraction sentence to answer the question.

The temperature at noon was 7.5° F above zero, but by 5 P.M. the temperature had fallen to 2.5° F below zero. What is the difference between the two temperatures?

6) –523 – (–276)

Mixed Review

7) Write four numbers that fall between 9.18 and 9.19.

8) Write 0.375 as an equivalent fraction (in simplest form).

9) Iodine and xenon are beside each other on the Periodic Table of elements. However, their melting points are very different. Iodine’s melting point is 113.7°C, while xenon’s melting point is –111.74°C. What is the difference between these two melting points? How could you use a number line to see if your answer is reasonable?

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Lesson 2.2.4 - Adding and Subtracting Rational Numbers

Active Instruction

ta(1) The vet put Desadra's dog, Cliff, on a diet. Cliff lost

ounces during month 1. He gained back

ounces in

month 2. During month 3, he lost

ounces, and another

ounces in month 4. What was the overall change to

Cliff's weight?

th(1) Write a numeric expression to answer each question. Solve and show your work.

The Quadratics' new song entered the music chart at #89. The song rose 23 places the first week, dropped 6 places the next week, and rose 12 places this week. What number on the chart is The Quadratics' song this week?

Team Mastery

(4) Write a numeric expression to answer each question. Solve and show your work.

Carey borrowed $1,250 from the bank for a business start-up loan. Each month for three months, Carey paid $100 toward the amount he owed. Then he borrowed $500 more to purchase business cards for his business. After that, he participated in a bank promotion to subtract $100 of debt on his loan for starting a checking account with the bank. What amount of money does Carey still owe to the bank?


Today your student added and subtracted rational numbers to solve real-world problems. Your student translated the situations into number sentences and then solved the problems. He or she used what they knew about adding and subtracting rational numbers with like and unlike signs to solve the problems.

Knowing how to add and subtract rational numbers is useful when calculating the temperature, elevations, status of stock market, personal budget, business accounts, or even the total yards gained or lost on a football field. In the next lesson, your student will use the order of operations to solve problems with multiple operations.

Your student should be able to answer the following questions about adding and subtracting rational numbers to solve real-world situations.

1) What are the key words in this situation? How will they help you write a numeric expression? 2) Which numbers will you combine first for this problem? 3) Describe what is happening in this situation.

Here are some ideas to work on adding and subtracting rational numbers to solve real-world situations: 1) Video: Adding/Subtracting Negative Numbers:

https://www.khanacademy.org/math/arithmetic/absolutevalue/adding_subtracting_negatives/v/adding-subtracting-negative-numbers

2) Activity: Battery Lab: http://illuminations.nctm.org/Lessons?PowerUp/PowerUp-AS-Voltmeter.pdf 3) Video: Subtract Rational Numbers in real-world context: http://learnzillion.com/lessons/1031subtract-

rational-numbers-in-realworld-contexts 4) Video: Adding and Subtracting Rational Numbers:

http://teachertube.com/viewVideo.php?title=Algebra_1_Section_2_5_Part_1&video_id=51352 5) Create three real-world situations that involve adding and subtracting rational numbers.

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Lesson Quick Look



Questions…


Today we added and subtracted rational numbers to solve real-world situations.

Knowing how to add and subtract rational numbers can help us budget our finances. For example, if you open a checking account with $50. Then you deposit $175 you made at a summer job. Would you have enough money in your account to purchase a new outfit that costs $250? If you did purchase the outfit, what would be the balance in your checking account?

First, write an expression for the problem. 50 + 175 – 250

Then, we combine like signs. 225 – 250

Let’s rewrite this using addition. 225 + (– 250)

Finally, we subtract. –25

If this was your checking account, you would not have enough money to purchase the outfit. If you did purchase the outfit, you would have a negative balance in your checking account; you would OWE the bank $25. It is important to keep track of your finances so you can avoid negative balances and going into debt.


1) Gracie fills her riding lawn mower with 2

gallons of gasoline. She uses gallon to mow

her front yard and 1 gallon to mow the back

yard. After finishing the back yard, Gracie, adds 1

gallon of gas. How much is in the tank now?

Explain your thinking.

2) Jonah downloaded 309.21 megabytes of games and 545.1 megabytes of music to his new smartphone. To save make room on his phone’s hard drive, he uploaded 216.87 megabytes of music and 184.27 megabytes of games to his computer and deleted them off his phone. How much space is he using on his smartphone for games and music?

3) Jonathan’s credit card balance was –$432.16. On Tuesday, he used his credit card to buy a new computer for $1,059.99. On Wednesday he

charged a new tax program for his computer for $75.99, and on Thursday he used his credit card to charge a new printer for $149.98. Jonathan plans to pay $95.50 on his bill. After his payment, how much debt will Jonathan have?

4) A SCUBA diving instructor has the class start their practice at 11 ft below the surface. He directs the student divers to ascend 9.2 feet, then descend 12.5 feet and wait for him to join them. Then the students descend 13.9 feet to take pictures of a parrot fish. Next the students ascend 5.7 feet to take pictures of a seahorse. At what depth are the divers now?

5) A California mountain, Telescope Peak, has an elevation of 11,049 feet. The mountain is near Death Valley, California which is –282 feet below sea level. What is the range of elevation between these two points?

Mixed Review

6) Draw a picture to find the difference.

–18 – 22

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Lesson 2.2.5 - Think Like a Mathematician: Build a Math Model I

Vocabulary

combinations permutations

Active Instruction

ta(1) Betty will make a pie using fruit available from local farms. She wants to use two different fruits in her pie. If she buys blueberries, strawberries, cherries, and peaches, how many different pies could she make?

ta(2) Cala, Li, and Sofia are contestants in a geography bee. How many ways can they be ordered for their turn to answer questions?

th(1) Jorge's snowcone stand sells pineapple, cherry, lemon-lime, and root beer flavored ices on Mondays. You can order either a kid-sized cup or an adult-sized cup. List the different ways you can order snowcones at Jorge's.

Team Mastery

(4) Rory, Valente, and Julienne are getting in line at the movie theater. Show the different ways they can stand in line.


Today your student learned how to solve problems by sorting and listing different options. Your student learned that combinations are ways to group items where order doesn’t matter, and that permutations are ways to group items where order does matter. Repeating items are not used for combinations, but they are used for permutations.

Your student should be able to answer these questions: 1. What is a combination? 2. What is a permutation? 3. How can you use an organized list for this problem? 4. Show me what a tree diagram looks like and how it’s used.

Here are some activities to try to practice listing and sorting at home:

1) Have your student pick out their favorite shirts and favorite shoes. Then have them write the number of different outfit combinations they could make.

2) Have your student plan activities to do for one afternoon. Then have them write the number of different orders in which they could do their activities.

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Lesson Quick Look



Questions…


Today, we sorted and listed different options to solve problems. We organized the different options to find all possible combinations and permutations. The options can be shown in an organized list or in a tree diagram. Here’s some examples!

Example 1: Mrs. Rawlings asked Ted, Ailine, Vida, and Cesar to form partner pairs to work on an assignment. How many ways can partner pairs be made?

Use an organized list to find all combinations of partners. Since order doesn’t matter, eliminate the repeating items in the list.

TA TV TC

AT AV AC

VT VA VC

CT CA CV

From the list, we can see that there are 6 combinations of partners.

Example 2: Mrs. Rawlings asked Ted, Ailine, and Vida to each explain a problem from their assignment. How many different orders can they explain their problem?

Use an organized list to find all permutations of the order they will explain their problem. Order does matter so we will keep the repeating items.

1st 2nd 3rd

From the list we can see that there are 6 orders in which the students can give their answers.


Directions for questions 1–10: Solve.

1) Murat has homework assignments for his social studies, English, and math classes. In how many different orders can he complete the assignments?

2) Nicolai purchased four dress shirts. One was white, one was blue, one was green, and one was pink. He also purchased four neckties. One was striped, one was checkered, one was solid, and one was polka-dot. List the ways Justin can match his dress shirts and neckties.

3) For lunch, a deli offers its customers small, medium, or large drinks. The customers can choose soda, unsweetened iced tea, or lemonade. List the different ways a drink can be ordered at the deli.

4) Petros’s Gyros sells gyros wrapped in either pita bread or in lettuce with different choices for meat or toppings. The meat choices are lamb, beef, and chicken. The topping choices are yogurt and hummus. How many different ways can a gyro be ordered at Petros’s?

Mixed Review 5) Find the difference. Show your work. –6.6 – 27.2

6) Find the sum. Show your work.

3 + (–4 ) + 8

Orders

T V

A

A V

T

V A

T

T

A

T

V

A

V

Documents

Unit 2 Cycle 1 - Adding and Subtracting Rational … | P a g e Unit 2 Cycle 1 - Adding and Subtracting Rational Numbers Lesson 2.1.1 - Definition of Rational Numbers Vocabulary integers