# 1.2 Operations With Rational Numbersmr. Mac's Page

MAC 1105 is a college level algebra course. This course introduces the student to the concept of functions and their graphs. Students will learn to graph linear, quadratic, rational, exponential, logarithmic, radical, power, and absolute value functions and transformations; perform operations with mathematical expressions and compositions of functions; find the inverse of a function; apply the. N-RN.1.1 “Explain definition of rational exponents from extending integer exponents.” N-RN.1.2 “Rewrite expressions involving radicals and rational exponents.” N-CN.1.1 “Know there is a complex number I and in the form of a + bi.” N-CN.1.2 “Operations with complex numbers.” A-REI.2.4 a-b “Solve quadratic equations.”. 6.use the order of operations with decimals, exponents, grouping symbols, arithmetic operations. 7.identify the order relation between two decimals or between a decimal and a fraction. IV.Concepts and skills associated with integers and rational numbers. 1.perform the four basic arithmetic operations on rational numbers. Apply properties of operations as strategies to multiply and divide rational numbers. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats. Solve real-world and mathematical problems involving the four operations with rational numbers. Rules for adding and subtracting rational numbers When adding numbers with the same signs, add the absolute value of each number and take the common sign.

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## Prime Time

Main Ideas Worked Homework Examples

In Prime Time students will explore certain important properties of whole numbers, especially those related to multiplication and division. The investigations in this unit will help students to:

- Understand relationships among factors, multiples, divisors, and products;
- Recognize and use properties of prime and composite numbers, even and odd numbers, and square numbers;
- Use rectangles to represent the factor pairs of numbers;
- Develop strategies for finding factors and multiples, least common multiples, and greatest common factors;
- Recognize and use the fact that every whole number can be written in exactly one way as a product of prime numbers;
- Use factors and multiples to solve problems and to explain some numerical facts of everyday life;
- Develop a variety of strategies for solving problems building models, making lists and tables, drawing diagrams, and solving simpler problems.

## Bits and Pieces I

Main Ideas Worked Homework Examples

In Bits and Pieces I students will investigate and make sense of rational numbers and operations with rational numbers. The investigations in this unit will help students to:

- Build an understanding of fractions, decimals, and percents and the relationships between and among these concepts and their representations;
- Develop ways to model situations involving fractions, decimals, and percents;
- Understand and use equivalent fractions to reason about situations;
- Compare and order fractions;
- Move flexibly between fraction, decimal, and percent representations;
- Use benchmarks such as 0, 1/2 , 1, and 1/4 to help estimate the size of a number or sum;
- Develop and use benchmarks that relate different forms of representations of rational numbers (for example, 50% is the same as 1/2 and 0.5);
- Use physical models and drawings to help reason about a situation;
- Look for patterns and describe how to continue the pattern;
- Use context to help reason about a situation;
- Use estimation to understand a situation.

## Shapes and Designs

Main Ideas Worked Homework Examples

Shapes and Designs was created to help students to:

- Understand some important properties of polygons and recognize polygonal shapes both in and out of the classroom;
- Investigate the symmetries of a shape-rotational or reflectional;
- Estimate the size of any angle using reference to a right angle and other benchmark angles;
- Use an angle ruler for making more accurate angle measurements;
- Explore parallel lines and angles created by lines intersecting parallel lines;
- Find patterns that help determine angle sums of polygons;
- Determine which polygons fit together to cover a flat surface and why;
- Explain the property of triangles that makes them useful as a stable structure for building;
- Reason about and solve problems involving shapes.

## Bits and Pieces II

Main Ideas Worked Homework Examples

Bits and Pieces II was created to help students make sense of one aspect of rational number: fraction operations. The investigations in this unit will help students to:

- Use benchmarks and other strategies to estimate the reasonableness of results of operations with fractions;
- Develop ways to model sums, differences, products, and quotients with areas, strips, and number lines;
- Use estimates and exact solutions to make decisions;
- Look for and generalize patterns in numbers;
- Use knowledge of fractions and equivalence of fractions to develop algorithms for adding, subtracting, multiplying and dividing fractions;
- Recognize when addition, subtraction, multiplication, and division is the appropriate operation to solve a problem;
- Write fact families to show the inverse relationship between addition and subtraction, and between multiplication and division;
- Solve problems using arithmetic operations on fractions;
- Look for and generalize patterns.

## Covering and Surrounding

Main Ideas Worked Homework Examples

In Covering and Surrounding student work with investigations leads them to:

- Understand area and relate area to covering a figure;
- Understand perimeter and relate perimeter to surrounding a figure;
- Develop strategies for finding areas and perimeters of rectangular shapes and non-rectangular shapes;
- Discover relationships between perimeter and area. including that each can vary while the other stays fixed;
- Understand how the areas of simple geometric figures relate to each other (e.g. the area of a parallelogram is twice the area of a triangle with the same base and height);
- Develop formulas and procedures-stated in words and/or symbols-for finding areas and perimeters of rectangles, parallelograms, triangles, and circles;
- Develop techniques for estimating the area and perimeter of an irregular figure;
- Recognize situations in which measuring perimeter or area will help answer practical questions.

## Bits and Pieces III

Main Ideas Worked Homework Examples

Bits and Pieces III, along with Bits and Pieces I and Bits and Pieces II, completes the development of meaning of, and operations on, fractions, decimals, and percents. This unit will help students to:

- Connect to what students already know about operations on fractions and whole numbers;
- Develop and use benchmarks and other strategies to estimate the answers to computations with decimals;
- Develop meaning of and algorithms for operations with decimals;
- Use the relationship between decimals and fractions to develop and understand why decimal algorithms work;
- Use the place value interpretation of decimals to make sense of short-cut algorithms for operations;
- Generalize number patterns to help make sense of decimal operations;
- Understand when addition, subtraction, multiplication or division is an appropriate operation to use to solve a problem;
- Understand that decimals are often associated with measurements in real world situations;
- Solve problems using operations on decimals;
- Use understanding of operations and the meaning of percents to solve percent problems of the form a% of b equals c for any one of the variables a, b, or c;
- Create and interpret circle graphs

## How Likely Is It?

Main Ideas Worked Homework Examples

The unit How Likely Is It? was created to help students:

- Understand that probabilities are useful for predicting what will happen over the long run
- Understand the concepts of equally likely and not equally likely;
- Understand that a game of chance is fair only if each player has the same chance of winning, not just a possible chance of winning;
- Understand that there are two ways to build probability models: by gathering data from experiments (experimental probability) and by analyzing the possible equally likely outcomes (theoretical probability);
- Understand that experimental probabilities are better estimates of theoretical probabilities when they are based on larger numbers of trials;
- Develop strategies for finding both experimental and theoretical probabilities;
- Critically interpret statements of probability to make decisions or answer questions.

## Data About Us

Main Ideas Worked Homework Examples

The unit Data About Us was created to help students:

- Understand and use the process of data investigation by posing questions, collecting and analyzing data distributions, and making interpretations to answer questions;
- Represent data distributions using line plots, bar graphs, stem-and-leaf plots, and coordinate graphs;
- Compute the mean, median, or mode and the range of the data;
- Distinguish between categorical data and numerical data and identify which graphs and statistics may be used to represent each kind of data;
- Make informed decisions about which graph or graphs and which of the measures of center (mean, median, or mode) and range may be used to describe a data distribution;
- Develop strategies for comparing data distributions

## Ruins of Montarek (CMP1)

Main Ideas Worked Homework Examples

The unit Ruins of Montarek was developed to help students:

- Read and create two-dimensional representations of three dimensional buildings
- Communicate spatial information
- Observe that the back view of a cube building is the mirror image of the front view and that the left view is the mirror image of the right view
- Understand and recognize line symmetry
- Explain how drawings of the base outline, front view and right view describe a building
- Construct cube buildings that fit two-dimensional building plans
- Develop a way to describe all buildings that can be made from a set of plans
- Understand that a set of plans can have more than one minimal building, but only one maximal building
- Explain how a cube can be represented on isometric dot paper, how the angles on the cube are represented by angles on the dot paper, and how the representations fit what the eye sees when viewing the corner of a cube building
- Make isometric drawings of cube buildings
- Visualize transformations of cube buildings, and make isometric drawings of the transformed buildings
- Reason about spatial relationships
- Use models and representations of models to solve problems

Write expressions that record operations with numbers and with letters standing for numbers.*For example, express the calculation 'Subtract y from 5' as 5 - y*.

Writing numeric and symbolic expressions and sentences to represent the operations required to solve problems is one of the most important skills in applied mathematics. Each unit of CMP3 includes many activities that develop understanding and proficiency in work on such modeling tasks.

In *Let’s be Rational*, there are several places where students are explicitly asked to model a situation with number sentences and equations: Problems 1.3, 1.4, 2.2, 3.1, 3.2, and 4.3. Here we present two of these Problems, 1.4 and 4.3, to demonstrate the unique ways that CMP3 builds students’ skills to model problems with number sentences and with equations containing variables.

*Let’s be Rational*** Problem 1.4**

*Let’s be Rational*

For example, using a Spice Recipe context, Problem 1.4 parts B, C, and D ask students explicitly to write number sentences to model problems of adding and subtraction spices. Then in part E, students use algebraic reasoning to interpret an existing equation and to write a story that would apply to that equation.

For Question B and C,

- Decide which operation you will use to solve each problem.
- Find an approximate answer using estimation.
- Write a number sentence and answer the question.

### B)

- How many ounces of spice does Betty buy?
- Tevin is allergic to cinnamon. If Betty removes cinnamon from the recipe for him, how many ounces of spice does she buy?

### C)

Ms. Garza buys spices to make one batch of Garam Masala. When she weighs her spices at home, she only has 10 ^{11}/_{12} ounces of spice. Which spice did Ms. Garza forget?

### D)

Renuka has two pounds of pepper in her cupboard. She knows that there are 16 ounces in one pound. After Renuka makes one batch of Garam Masala, how many ounces of pepper does Renuka have left in her cupboard?

### E)

For each number sentence below, write a spice story. Then find the value for *N* that makes the sentence true.

- 3
^{1}/_{6}- 1^{3}/_{4}=*N* *N*+^{3}/_{4}= 1^{1}/_{2}- 2
^{2}/_{3}-*N*= 1^{1}/_{4}

*Let’s be Rational* Problem 4.3

In Problem 4.3 of *Let’s Be Rational* asks students to analyze five different word problem situations, to decide which operations are needed to solve the problems, and to write one or more expressions that represent those operations.

The exercises involve writing expressions to record problem-solving operations in* Let’s Be Rational* and result in number sentences like the following, which applies to Part C:

Sasha bought 3 1/2 pints of blueberries to make jelly. She ate 3/4 of a pint of berries on her way home. How many pints of berries does she have left to make jelly?

3 ^{1}/_{2} - ^{2}/_{4} = 2 ^{3}/_{4} [*Let's Be Rational *4.3C]

Understandings and skills used in writing numeric expressions and equations are extended to use of letter names for variables in several other problems of *Let’s Be Rational.* For example, Problem 4.3H:

**H** A grandmother is making clothes for her three granddaughters. She will make a jacket and one other item for each granddaughter. The three other items will be exactly the same. A jacket takes 1 ^{5}/_{8} yards of fabric. She has ten yards of material in all. She is trying to figure out how much fabric she has for each of the three extra items.

Let *N* represent the fabric needed one extra item. Explain why each of the sentences below does or does not describe the situation. (More than one sentence may apply.)

- 3(1
^{5}/_{8}+*N*) = 10 - 10 - 4
^{7}/_{8}= 3*N* - 3 x 1
^{5}/_{8}+*N*= 10 - 10 ÷ 1
^{5}/_{8}=*N*

This problem is answered by the equations 3(1 ^{5}/_{8}+*N*)=10 and 10-4 ^{7}/_{8}=3*N*.

Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. *For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms*.

The mathematical terms *sum, product, factor *and* quotient* are used throughout *Let’s Be Rational *unit*. *The term *quotient* is defined and used in development of fraction division algorithms in *Let’s Be Rational *Investigation 3.

## Investigation 3 Dividing with Fractions

So far in *Let's Be Reational*, you have solved problems using addition, subtraction and multiplication. In Investigation 3, you will solve problems that require division of fractions. As you work on these problems, think about similarities and differences among the problems.

In the number sentence 21 ÷ 7 = 3, 21 is *dividend*, 7 is the *divisor*, and 3 is the result, or *quotient*.

You can use the vocabulary of division problems as placeholders in a division number sentence. The division sentence below shows how these quantities relate to one another.

dividend ÷ divisor = quotient

First, you need to understand what division of fractions means. Then you can calculate quotients when the divisor or the dividend, or both, is a fraction.

Apply the properties of operations to generate equivalent expressions.*For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y*.

## Let's be Rational Problem 2.3A

The idea of equivalence comes up throughout CMP3 implicitly during the Summary of lessons as students share equivalent methods to solve problems and their teachers help them to make connections between those problem solutions. CMP3 also explicitly asks students to compare solution strategies in *Let’s be Rational* Problem 2.3. In each of these strategies, the steps involved in solving the problem are equivalent expressions.

## Problem 2.3

### A)

- Takoda and Yuri are computing
^{1}/_{2}x 2^{2}/_{3}. What is a reasonable estimate for this product? - Takoda and Yuri each use a different strategy.
#### Takoda's Strategy

I used what I know about fractions to rewrite 2

^{2}/_{3}as^{8}/_{3}to make the problem easier to solve.^{1}/_{2}x 2^{2}/_{3}=^{1}/_{2}x^{8}/_{3}

=^{8}/_{6}

= 1^{2}/_{6}

= 1^{1}/_{3}#### Yuri's Strategy

I wrote 2

^{2}/_{3}as (2 +^{2}/_{3}) and used the Distributive Property to make the problem easier to solve.^{1}/_{2}x 2^{2}/_{3}=^{1}/_{2}x (2 +^{2}/_{3})

= (^{1}/_{2}x 2) + (^{1}/_{2}x^{2}/_{3})

= 1 +^{2}/_{6}

= 1^{2}/_{6}

= 1^{1}/_{3}- Does each strategy work? How do you know?
- How are the strategies similar? How are they different?

3. Use both strategies to solve 1 ^{1}/_{3} x ^{4}/_{5}. Then check your answer with a drawing.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).*For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.*

*Let’s be Rational *Problem 2.3A

The idea of equivalence comes up again in considering equivalence of fractions and ratios in the *Let’s Be Rational* unit. For instance, Problem 2.3 (see above) encourages students to compare two strategies for simplifying the multiplication of a fraction and a mixed number. The steps in each solution method are equivalent number sentences.

*Let’s be Rational*** Problem 4.3**

*Let’s be Rational*

Understandings and skills used in writing numeric expressions and equations are extended to use of letter names for variables in several other problems of *Let’s Be Rational.* In Problem 4.3H (see above), this idea is extended to showing how two equations are equivalent representations of a single problem solution.

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

*Let’s be Rational*** Problem 1.2**

*Let’s be Rational*

The general idea of solving inequalities is first introduced informally in Problem 1.2 in *Lets’ Be Rational*. This informal introduction is anchored in the context of reasoning about overestimates and underestimates.

Sometimes you should**overestimate,**or give an estimate that is a bit bigger than the actual value. Overestimate to make sure you have enough. Sometimes you should**underestimate,**or give an estimate that is a bit smaller than the actual value. Underestimate to stay below a certain limit.

For Question A-E,

- Answer the question by using estimation. Explain your reasoning.
- Explain how confident you are in your answer.
- For each estimate you make, tell whether it is an overestimate or an underestimate. Explain why you chose to overestimate or underestimate.

**A)** Mrs. Edwards is building a dollhouse for her children. She needs to buy wood for the railing on the balcony.

The wood is available in 12-inch, 14-inch, and 16-inch lengths. She does not want to waste wood. What length should she buy?

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

*Let’s be Rational*** Problem 1.4E**

*Let’s be Rational*

In Problem 1.4E students are asked to interpret variable equations and write a contextual story that matches the equation.

**E)** For each number sentence below, write a spice story. Then find the value for *N* that makes the sentence true.

- 3
^{1}/_{6}- 1^{3}/_{4}= N - N +
^{3}/_{4}= 1^{1}/_{2} - 2
^{2}/_{3}- N = 1^{1}/_{4}

*Let’s be Rational*** Investigation 4**

*Let’s be Rational*

Students’ confidence and ability to work with expressions and equations builds throughout the unit. In Investigation 4 of *Let’s be Rational*, students use variables to represent numbers and write expressions to solve mathematical and real world problems.This is elaborated further below.

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

*Let’s Be Rational*** Investigation 4**

*Let’s Be Rational*

Several CMP3 units develop student understanding and skill in work with these equations. The key idea in those earlier encounters with equation solving is *fact families*. For any three numbers *a, b,* and *c*, if *a* + *b* = *c*, then we can infer that *b* + *a* = *c*, *a* = *c* –* b, *and *b* = *c* – *b*. Similarly, for any three numbers *a, b,* and *c*, if *ab* = *c* we can infer that *ba *= *c*, *a* = *c *÷ *b (b ≠ *0) and *b* = *c *÷ a (*a* ≠ 0). These relationships can be used to solve any one-step linear equation.

For example, to solve*n*+ 5 = 34 we reason as follows:

If *n *+ 5 = 34, then *n *= 34 – 5 or *n *= 29

Checking, we see that 29 + 5 = 35.

Similarly, to solve 5*n* = 35 we reason as follows:

If 5*n* = 35, then *n *= 35 ÷ 5 or *n *= 7

Checking, we see that 5(7) = 35.

This use of fact families to solve one-step linear equations is developed and practiced in the *Let’s Be Rational* unit. So students should have efficient and easy to recall strategies for solving *x *+ *p *= *q *and *px *= *q *before they get to *Variables and Patterns.*

*Let’s be Rational*** Problem 4.1**

*Let’s be Rational*

Below are two fact families expressing relationships among fractions.

Example 1 | Example 2 | |
---|---|---|

Addition Sentence | ^{3}/_{4} ÷ ^{1}/_{8} = ^{7}/_{8} | ^{6}/_{8} + N = ^{7}/_{8} |

Related Number Sentences | ^{1}/_{8} ÷ ^{3}/_{4} = ^{7}/_{8}^{7}/_{8} - ^{3}/_{4} = ^{1}/_{8}^{7}/_{8} - ^{1}/_{8} = ^{3}/_{4} | N + ^{6}/_{8} = ^{7}/_{8}^{7}/_{8} - ^{6}/_{8} = N^{7}/_{8} - N = ^{6}/_{8} |

- How are these fact families different from each other?
- How do you know that the three related sentences below the top row are true?

In Problem 4.1 you will create fact families and use them to find unknown numbers.

**A)** For each number sentence, write a complete fact family and find the value of *N*.

^{5}/_{10}-^{2}/_{5}=*N*- 3
^{3}/_{5}+ 1^{2}/_{3}=*N* - Describe the relationship between addition and subtraction. Use the fact families in parts (1) and (2) as examples.

**B)** For each number sentence, find the value *N*.

*N*+ 1^{2}/_{3}= 5^{5}/_{6}^{3}/_{4}+*N*=^{17}/_{12}*N*-^{1}/_{2}=^{3}/_{8}- How can fact families help you find the value of N in parts (1) - (3)?

*Let’s be Rational*** Problem 4.2**

*Let’s be Rational*

Now it is your turn to write fact families.

### 1.2 Operations With Rational Numbersmr. Mac's Page Key

**A)**Write a complete fact family for each of the following sentences.

^{2}/_{3}×^{1}/_{5}=^{2}/_{15}^{3}/_{4}×^{5}/_{8}=^{15}/_{32}^{9}/_{40}÷^{3}/_{5}=^{3}/_{8}^{4}/_{15}÷^{2}/_{5}=^{2}/_{3}

**B)** Write a complete fact family for each of the following equations. Use your fact family to find the value of *N*

^{3}/_{8}×*N*=^{21}/_{80}^{2}/_{3}×*N*=^{10}/_{15}- 1 ÷
*N*=^{2}/_{3} ^{8}/_{15}÷*N*=^{2}/_{3}

**C)**Marla says she can use the idea of *decomposing* a product to find the unknown factor, *N*, in 15 = 2 × N. She rearranges this multiplication sentence as the division sentences 15 ÷ 2 = *N*. Does Marla's strategy work? Explain why or why not?

### 1.2 Operations With Rational Numbersmr. Mac's Page Numbers

**D)** Below are sets of three numbers. Some of these sets can be related using addition and subtraction. Some can be related using multiplication and division. For each set, identify what the relation is. Then write a complete fact family.

^{3}/_{5},^{1}/_{3},^{14}/_{15}^{3}/_{4},^{4}/_{3}, 1- 1
^{1}/_{2}, 2^{2}/_{3}, 4^{1}/_{6} ^{3}/_{2}, 3,^{9}/_{2}

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Thorough and explicit attention to solving inequality equations does not formally occur until *Variables and Patterns*. If there is a need to introduce inequalities earlier than that, problems in* Let’s be Rational* can be extended to include problems with solutions that can be written using an inequality.

*Let’s be Rational*** Problems 1.2 and 4.3**

*Let’s be Rational*

### 1.2 Operations With Rational Numbersmr. Mac's Page Shortcut

At the end of* Let’s be Rational,* following Investigation 4, you may want to revisit the overestimation and underestimation ideas from Problem 1.2 and rewrite these situations as inequalities. Here we present an example using Problem 1.2D, of how this might be accomplished.

**D)** Jasmine's raspberry jam recipe calls for 4 ^{1}/_{3} quarts of raspberries. She has picked 3 ^{1}/_{2} quarts of raspberries. About how many more quarts of raspberries should she pick?

The following scenario is an example of how this could be revisited and extended to explicitly include solving an inequality:

### 1.2 Operations With Rational Numbersmr. Mac's Page Number

Teacher: “When we solved this problem, we said that this was an overestimate. Why is that?”

Student: “Because she needs *at least* 4 ^{1}/_{3}quarts of raspberries. If she picks more than that, it is OK, she will just have a little left over. But she cannot pick less than that and still have enough to make her jam.'

Teacher: “How might we write this using an inequality statement?”

Student: 3 ^{1}/_{2} + *N* ≥ 4 ^{1}/_{3}