problem solving analyze relationships lesson 9 3

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Unit 1: Module 1: Relationships between quantities and reasoning with equations and their graphs

About this unit.

"In this module students analyze and explain precisely the process of solving an equation. Through repeated reasoning, students develop fluency in writing, interpreting, and translating between various forms of linear equations and inequalities and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations. They learn the terminology specific to polynomials and understand that polynomials form a system analogous to the integers." Eureka Math/EngageNY (c) 2015 GreatMinds.org

Topic A: Lessons 1-3: Piecewise, quadratic, and exponential functions

• Introduction to piecewise functions (Opens a modal)
• Worked example: evaluating piecewise functions (Opens a modal)
• Parabolas intro (Opens a modal)
• Intro to exponential functions (Opens a modal)
• Exponential vs. linear growth (Opens a modal)
• Exponential vs. linear models: verbal (Opens a modal)
• Exponential vs. linear models: table (Opens a modal)
• Evaluate piecewise functions Get 3 of 4 questions to level up!
• Evaluate step functions Get 3 of 4 questions to level up!
• Parabolas intro Get 3 of 4 questions to level up!
• Warmup: exponential vs. linear growth Get 3 of 3 questions to level up!
• Exponential vs. linear growth Get 3 of 4 questions to level up!
• Exponential vs. linear models Get 3 of 4 questions to level up!

Topic A: Lessons 4-5: Analyzing graphs

• Graph interpretation word problem: temperature (Opens a modal)
• Graph interpretation word problem: basketball (Opens a modal)
• Linear graphs word problems (Opens a modal)
• Linear graphs word problem: cats (Opens a modal)
• System of equations word problem: walk & ride (Opens a modal)
• Graph interpretation word problems Get 3 of 4 questions to level up!
• Linear equations word problems: graphs Get 3 of 4 questions to level up!
• Systems of equations word problems Get 3 of 4 questions to level up!

Topic B: Lesson 8: Adding and subtracting polynomials

• Polynomials intro (Opens a modal)
• The parts of polynomial expressions (Opens a modal)
• Evaluating polynomials (Opens a modal)
• Simplifying polynomials (Opens a modal)
• Adding polynomials (Opens a modal)
• Subtracting polynomials (Opens a modal)
• Polynomial subtraction (Opens a modal)
• Adding & subtracting multiple polynomials (Opens a modal)
• Polynomials intro Get 3 of 4 questions to level up!
• Add polynomials (intro) Get 3 of 4 questions to level up!
• Subtract polynomials (intro) Get 3 of 4 questions to level up!
• Add & subtract polynomials Get 3 of 4 questions to level up!

Topic B: Lesson 8: Adding and subtracting polynomials with 2 variables

• Adding polynomials: two variables (intro) (Opens a modal)
• Subtracting polynomials: two variables (intro) (Opens a modal)
• Subtracting polynomials: two variables (Opens a modal)
• Finding an error in polynomial subtraction (Opens a modal)
• Add & subtract polynomials: two variables (intro) Get 3 of 4 questions to level up!
• Add & subtract polynomials: two variables Get 3 of 4 questions to level up!
• Add & subtract polynomials: find the error Get 3 of 4 questions to level up!

Topic B: Lesson 9: Multiplying monomials

• Multiplying monomials (Opens a modal)
• Multiplying monomials to find area: two variables (Opens a modal)
• Multiplying monomials to find area (Opens a modal)
• Multiplying monomials challenge (Opens a modal)
• Multiply monomials Get 3 of 4 questions to level up!
• Multiply monomials (advanced) Get 3 of 4 questions to level up!

Topic B: Lesson 9: Multiplying monomials by polynomials

• Multiplying monomials by polynomials: area model (Opens a modal)
• Multiplying monomials by polynomials (Opens a modal)
• Multiplying monomials by polynomials challenge (Opens a modal)
• Multiply monomials by polynomials: area model Get 3 of 4 questions to level up!
• Multiply monomials by polynomials Get 3 of 4 questions to level up!
• Multiply monomials by polynomials challenge Get 3 of 4 questions to level up!

Topic B: Lesson 9: Multiplying binomials

• Multiplying binomials: area model (Opens a modal)
• Multiplying binomials intro (Opens a modal)
• Multiplying binomials (Opens a modal)
• Multiply binomials: area model Get 3 of 4 questions to level up!
• Warmup: multiplying binomials intro Get 3 of 4 questions to level up!
• Multiply binomials intro Get 3 of 4 questions to level up!
• Multiply binomials Get 3 of 4 questions to level up!

Topic C: Lessons 10-13: Solving Equations

• Testing solutions to equations (Opens a modal)
• Intro to equations (Opens a modal)
• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Same thing to both sides of equations (Opens a modal)
• Representing a relationship with an equation (Opens a modal)
• Dividing both sides of an equation (Opens a modal)
• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
• Testing solutions to equations Get 5 of 7 questions to level up!
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Topic C: Lesson 14: Solving inequalities

• Testing solutions to inequalities (Opens a modal)
• Plotting inequalities (Opens a modal)
• Plotting an inequality example (Opens a modal)
• One-step inequalities examples (Opens a modal)
• One-step inequalities: -5c ≤ 15 (Opens a modal)
• Two-step inequalities (Opens a modal)
• Inequalities with variables on both sides (Opens a modal)
• Inequalities with variables on both sides (with parentheses) (Opens a modal)
• Multi-step inequalities (Opens a modal)
• Testing solutions to inequalities Get 3 of 4 questions to level up!
• Plotting inequalities Get 3 of 4 questions to level up!
• One-step inequalities Get 5 of 7 questions to level up!
• Two-step inequalities Get 5 of 7 questions to level up!
• Multi-step linear inequalities Get 3 of 4 questions to level up!

Topic C: Lessons 15-16 Compound inequalities

• Compound inequalities: OR (Opens a modal)
• Compound inequalities: AND (Opens a modal)
• A compound inequality with no solution (Opens a modal)
• Double inequalities (Opens a modal)
• Compound inequalities examples (Opens a modal)
• Compound inequalities Get 3 of 4 questions to level up!

Topic C: Lessons 17-19: Advanced equations

• Zero product property (Opens a modal)
• Solving quadratics by factoring (Opens a modal)
• Rational equations intro (Opens a modal)
• Linear equations with unknown coefficients (Opens a modal)
• Zero product property Get 3 of 4 questions to level up!
• Quadratics by factoring (intro) Get 3 of 4 questions to level up!
• Rational equations intro Get 3 of 4 questions to level up!
• Linear equations with unknown coefficients Get 3 of 4 questions to level up!

Topic C: Lesson 20: Solution sets to equations with two variables

• Two-variable linear equations intro (Opens a modal)
• Solutions to 2-variable equations (Opens a modal)
• Worked example: solutions to 2-variable equations (Opens a modal)
• Completing solutions to 2-variable equations (Opens a modal)
• Solutions to 2-variable equations Get 3 of 4 questions to level up!
• Complete solutions to 2-variable equations Get 3 of 4 questions to level up!

Topic C: Lesson 21: Solution sets to inequalities with two variables

• Intro to graphing two-variable inequalities (Opens a modal)
• Graphing two-variable inequalities (Opens a modal)
• Two-variable inequalities from their graphs (Opens a modal)
• Solutions of inequalities: algebraic Get 3 of 4 questions to level up!
• Solutions of inequalities: graphical Get 3 of 4 questions to level up!
• Graphs of inequalities Get 3 of 4 questions to level up!
• Two-variable inequalities from their graphs Get 3 of 4 questions to level up!

Topic C: Lesson 22: Solution sets to simultaneous equations

• Testing a solution to a system of equations (Opens a modal)
• Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 (Opens a modal)
• Systems of equations with graphing: exact & approximate solutions (Opens a modal)
• Testing solutions to systems of inequalities (Opens a modal)
• Intro to graphing systems of inequalities (Opens a modal)
• Graphing systems of inequalities (Opens a modal)
• Solutions of systems of equations Get 3 of 4 questions to level up!
• Systems of equations with graphing Get 3 of 4 questions to level up!
• Solutions of systems of inequalities Get 3 of 4 questions to level up!
• Systems of inequalities graphs Get 3 of 4 questions to level up!

Topic C: Lesson 23: Solution sets to simultaneous equations

• Systems of equations with elimination: King's cupcakes (Opens a modal)
• Systems of equations with elimination: x-4y=-18 & -x+3y=11 (Opens a modal)
• Systems of equations with elimination: potato chips (Opens a modal)
• Systems of equations with elimination (and manipulation) (Opens a modal)
• Why can we subtract one equation from the other in a system of equations? (Opens a modal)
• Worked example: equivalent systems of equations (Opens a modal)
• Worked example: non-equivalent systems of equations (Opens a modal)
• Systems of equations with elimination Get 3 of 4 questions to level up!
• Systems of equations with elimination challenge Get 3 of 4 questions to level up!
• Reasoning with systems of equations Get 3 of 4 questions to level up!

Topic C: Lesson 24: Applications of systems of equations and inequalities

• Age word problem: Imran (Opens a modal)
• Age word problem: Ben & William (Opens a modal)
• Age word problem: Arman & Diya (Opens a modal)
• Modeling with systems of inequalities (Opens a modal)
• Writing systems of inequalities word problem (Opens a modal)
• Solving systems of inequalities word problem (Opens a modal)
• Graphs of systems of inequalities word problem (Opens a modal)
• Age word problems Get 3 of 4 questions to level up!
• Systems of inequalities word problems Get 3 of 4 questions to level up!

Topic D: Creating equations to solve problems

• Worked example: Rate problem (Opens a modal)
• Sequences intro (Opens a modal)
• Evaluating sequences in recursive form (Opens a modal)
• Rate problems 2 Get 3 of 4 questions to level up!
• Evaluate sequences in recursive form Get 3 of 4 questions to level up!

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• 6th Grade Mathematics

Education Standards

Maryland college and career ready math standards.

Learning Domain: Expressions and Equations

Standard: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Common Core State Standards Math

Cluster: Represent and analyze quantitative relationships between dependent and independent variables

Reviewing Quantitative Relationships

In this lesson, students represent quantitative relationships involving rates using tables, graphs, double number lines, and formulas. Students will understand how to create one such representation when given another representation.

Key Concepts

Quantitative relationships involving rates can be represented using tables, graphs, double number lines, and formulas. One such representation can be used to create another representation. Two rates can describe each situation: the rate and its inverse. For the water pump situation, there are two related formulas: a formula for finding the quantity of water pumped for any amount of time, and a formula for finding the amount of time for any quantity of water.

Goals and Learning Objectives

• Understand that tables, graphs, double number lines, and formulas can be used to represent the same situation.
• Compare the different representations within a situation and the same representation across similar situations.
• Understand each representation and how to find the rate in each one.

Graph of Pumping Water

Lesson guide.

Review the graph and make sure students know how to read the coordinates and interpret the graph.

• The point (10,200) tells you 200 gallons were pumped in 10 seconds.
• The rate is 20 gallons/second.

The graph shows the amount of water pumped over time.

• What does the point (10, 200) tell you?

Math Mission

Discuss the Math Mission. Students will represent quantitative relationships in different ways.

SWD: Students with disabilities may struggle to independently find patterns and to observe similarities between formulas. Use your understanding of your students' needs to inform any pre-teaching and/or interventions you may provide.

Represent rates in different ways.

Graph the Relationships

As students work, look for these types of responses:

• Students that find the rate by solving for gallons per minute.
• Students that find the rate by solving for minutes per gallon.
• Students who use different types of formulas for the situations, because the inconsistency will cause problems when graphing.

SWD: Color coding of ideas, relationships, and formulas can be helpful for students with special needs. Consider providing these color-coded formulas to students once they have learned about the concepts.

Mathematical Practices

Mathematical Practice 6: Attend to precision; and Mathematical Practice 1: Make sense of problems and persevere in solving them.

• To find each rate and formula, students must first make sense of the problem situation and attend to precision. In some situations, the rate is given, and students need to write the formula. In other situations, students need to calculate the rate. In some situations, the time is given first and then the number of gallons. In other situations, the number of gallons is given first and then the time.
• Later, students will compare the different situations on the same graph. While graphing, students may have difficulty selecting axes value ranges that will work for all four situations. Identify student work that has different values for the axes and that attends to different levels of precision.

Mathematical Practice 4: Model with mathematics.

• Students will be working from one model in order to create another model of the same situation (for example, from a table to a formula to a graph). Watch for students who are able to model the mathematical relationships correctly.
• Also look for students who are able to use their graph to draw conclusions about the four situations.

Interventions

Student is having difficulty getting started.

• Describe what you need to do for each situation to your partner.
• What information do you know?
• What is the first thing you need to find?
• Once you identify the rate, what will you do?
• How can you use your formula to make a graph of the situation?

Student doesn't understand how to write a formula.

• How do you know your formula is correct?
• What do the variables in your formula represent?
• What formula matches this line on the graph? Explain how you know.
• How are they the same?
• How are they different?
• Can you use both formulas to make the same graph?

Make sure your students are correctly representing the units.

• What does the vertical axis of your graph show? What is the scale? What size intervals did you use?
• What does the horizontal axis of your graph show? What is the scale? What size intervals did you use?
• Does the range of your scales cover all problem situations?
• Pump 1: g = 125 t
• Pump 2: g = 52 t
• Pump 3: g = 260 t
• Pump 4: g = 52 t
• Pump 1: t = (1,125) g
• Pump 2: t = (152) g
• Pump 3: t = (1,260) g
• Pump 4: t = (152) g

Below are four situations describing the rate of water flowing through a pump.

Situation 1: Pump 1 pumps water at a rate of 125 gallons per minute.

Situation 2: Pump 2 pumps water at a constant rate. Pump 2 pumps 364 gallons in 7 minutes.

Situation 3: Pump 3 pumps at a constant rate, represented by the table.

Situation 4: Pump 4 pumps at a constant rate, represented by the double number line.

For each situation, do the following.

• Write a formula that represents the rate relationship.
• Graph all four relationships on a single graph.

Situation 4:

• What rate will you find gallons per minute or minutes per gallon?
• How will you represent this rate as a formula?
• Why should you represent time on the x -axis?

Prepare a Presentation

Preparing for ways of thinking.

• Look for students who wrote equations that solve for gallons, in particular students whose methods may differ from others.
• Look for students who wrote equations that solve for minutes, in particular students whose methods may differ from the others.
• Look for students who had issues with graphing due to the formula they wrote.
• Look for students who have a clear understanding of the relationship between rate and the slope of the graph.

Challenge Problem

• Pump 1: g = 125 t or t = ( 1 125 ) g
• Pump 2: g = 52 t or t = ( 1 52 ) g
• Pump 3: g = 260 t or t = ( 1 260 ) g
• Pump 4: g = 52 t or t = ( 1 52 ) g
• Possible answer: We are interested in how many gallons of water there will be after a given time, so the number of gallons depends on the time. Thus, time is the independent variable and gallons is the dependent variable. The formula g = kt indicates that time is the independent variable and g is the dependent variable. The formula t = kg indicates that volume is the independent variable and time is the dependent variable.

Prepare a presentation about the four situations.

• Compare the situations. What is similar, and what is different?
• What did you notice about two of the situations?
• Which line on your graph of the rate relationships is the steepest? Explain why.
• In this problem, the time is considered the independent variable and the volume the dependent variable. Explain why.

Make Connections

By the end of the discussion, students should understand each others' methods and make connections between methods.

If any students work on the challenge problems, have them share their thinking about independent and dependent variables.

ELL: When discussing how to find the formulas, make it a point to write, in words, what each of the formulas represents. It is important to provide students with the opportunity to practice a combination of written and oral skills.

Mathematics

Make note of any mathematical practices that you observe students using.

Ask the following questions:

• What is similar?
• What is different?
• Which graph is easiest to read?
• Did anyone use different strategies? Explain what you did.
• Where do you see the rate in each representation (table, double number line, graph, formula)?
• Which representation of the rate situation is easiest for you to understand? Why?
• Which pump is the fastest? How do you know?
• Which pump is the slowest? How do you know?
• How does the graph show that each pump works at a constant rate?

Performance Task

Ways of thinking: make connections.

Take notes as you listen to your classmates’ solutions.

As your classmates present, ask questions such as the following:

• What strategies did you use to make sense of the four situations?
• Whose method makes most sense to you? Why?
• What conclusions can you draw from interpreting the graph?

Summarize Your Learning

A possible summary.

You can represent rate situations using tables, double number lines, graphs, or formulas. You can use one representation to create another representation.

Additional Discussion Points

• Quantitative relationships involving rates can be represented using tables, graphs, double number lines, and formulas. One such representation can be used to create another representation.
• Two rates can describe each situation: each rate is the inverse of the other.
• For the water pump situation, there are two related formulas: a formula for finding the quantity of water pumped for any amount of time and a formula for finding the amount of time for any quantity of water.

Formative Assessment

Summary of the math: summarize your learning.

Summarize the mathematics you learned about graphing rates.

Check your summary.

• Does your summary explain what each point on your graph represents?
• Does your summary include the term constant rate ?
• Does your summary describe the two rates for each situation and their related formulas?

Check Your Understanding

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery problems and which students would benefit from review before the assessment. Have students work on the Self Check individually.

Have students submit their work to you. Make notes about what their work reveals about their current levels of understanding and their different problem-solving approaches. Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

ELL: It might be  helpful for ELLs and other students to have a sample or model for the concepts, strategies, and applications that will be addressed in the quiz and the format you want them to follow. Be prepared to address and explicitly re-teach or review vocabulary, concepts, strategies, and applications.

SWD: Students with disabilities may benefit from having the Self Check presented in a variety of ways (as auditory and visual information). Provide students the option of listening to the content for this task. Some students with disabilities may need additional time to complete the Self Check. Be sure to make provisions for the additional time or consider reducing the number of tasks required for students to demonstrate mastery of skill.

Student has trouble getting started

• The scale of the graph is in inches; which animal is more likely to travel just one inch at a time?

Student does not know how to write a formula.

• Look at the formulas you wrote in the previous lessons. How do those formulas relate to this situation?
• What is the general formula to find distance? How can you use this to help you write the formula for this situation?
• What is the general formula to find time? How can you use this to help you write the formula for this situation?
• The graph most likely shows the speed of a garden snail.
• In 8 minutes, the creature traveled 0.4 inch.
• In 3 minutes, the creature traveled 0.15 inch.
• The creature's rate of speed is 0.05 inch per minute.
• d = 12   min ⋅ 0.05 in. min = 0.6   in. In 12 minutes, the creature travels 0.6 inch.
• t = 2   in. ÷ 0.05 in. min = 40   min . It takes the creature 40 minutes to travel 2 inches.

Use the graph to answer the questions.

• Does this graph most likely show the speed of a garden snail or a cheetah?
• How far did this creature travel in 8 minutes?
• How far did this creature travel in 3 minutes?
• What is this creature’s rate of speed in inches per minute?
• Solve an equation to find the distance the creature travels in 12 minutes.

Reflect On Your Work

Have each student do a quick reflection before the end of class. Review the reflections.

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

One thing I know about representing rate relationships is …

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Eureka Math Grade 3 Module 6 Lesson 9 Answer Key

Engage ny eureka math 3rd grade module 6 lesson 9 answer key, eureka math grade 3 module 6 lesson 9 pattern sheet answer key.

Eureka Math Grade 3 Module 6 Lesson 9 Problem Set Answer Key

Question 1. Four children went apple picking. The chart shows the number of apples the children picked.

Question 2. Use the chart or graph to answer the following questions. a. How many more apples did Stewart and Roxanne pick than Philip and Trisha? b. Trisha and Stewart combine their apples to make apples pies. Each pie takes 7 apples. How many pies can they make? Answer: a. eight more apples Stewart and Roxanne pick than Philip and Trisha b. Trisha and Stewart combine their apples to make apples pies. Each pie takes 7 apples.  they can make are 5 pies

Question 3. Ms. Pacho’s science class measured the lengths of blades of grass from their school field to the nearest \(\frac{1}{4}\) inch. The lengths are shown below.

Eureka Math Grade 3 Module 6 Lesson 9 Exit Ticket Answer Key

Eureka Math Grade 3 Module 6 Lesson 9 Homework Answer Key

Question 1. The table below shows the amount of money Danielle saves for four months.

Question 2. 2. Use the table or graph to answer the following questions. a. How much money does Danielle save in four months? b. How much more money does Danielle save in March and April than in January and February? c. Danielle combines her savings from March and April to buy books for her friends. Each book costs $9. How many books can she buy? d. Danielle earns $33 in June. She buys a necklace for $8 and a birthday present for her brother. She saves the $13 she has left. How much does the birthday present cost? Answer: a.$90 Explination: $9+$18+$36+$27 =$90 b. $36 more Explination: Total amount saved on March and April combined = $9+$18 =$27 Total amount saved on January  and February combined = $36+$27 =$63 $63-$27 =$36 c. 3 books Explination: Given: Savings of March and April = $27 One book cost = $9 Therefor $9+$9+$9=$27 d. $12 Explination: Given: Necklace cost = $8 Saved amount = $13 Total earning of june =$33 There by $8+$13+x=$33 $21+x=$33 x=$33-$21 x=$12

Eureka Math Grade 3 Module 6 Answer Key

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Chapter 9, Lesson 3: Angle Relationships

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