Integrated Math 1 Module 1 Equations and Inequalities

Transcription

1 1 Integrated Math 1 Module 1 Equations and Inequalities Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius In partnership with the Utah State Office of Education 2012 Mathematics Vision Project MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution NonCommercial ShareAlike 3.0 Unported license

2 2 Module 1 Overview Prerequisite Concepts & Skills: Operations with integers, fractions, decimals and variable expressions Solving basic one and two step equations and inequalities Understanding of the number line and the coordinate plane Plotting points Evaluating expressions Order of operations Summary of the Concepts & Skills in Module 1: Team building skills and group roles Communication skills (orally & written) Introduction to CC Standards of Math Practices through daily tasks Combine like terms Use the distributive property Write expressions to represent a context and/or given a visual Solve linear equations Solve linear inequalities & graphing solutions on a number line Solve absolute value equations and inequalities Solve literal equations Convert between slope-intercept and standard form of linear functions Graph lines in slope-intercept and standard form Find slope between two coordinate pairs Write linear equation and inequalities to represent a context Content Standards and Standards of Mathematical Practice Covered: Content Standards: N.Q.1, N.Q.2, A.SSE.1, A.CED.1, A.CED.2, A.CED.4, A.REI.1, A.REI.3, A.REI.3.1 CA Standards of Mathematical Practice: 1. Make sense of problems & persevere in solving them. 2. Reason abstractly & quantitatively 3. Construct viable arguments & critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Materials: Graph paper Graphing utility (graphing calculator, Desmos, Geogebra, etc.) Colored pens/pencils Cups and Coins manipulatives-dixie cups, small bags/clear containers, and coins/counters Paper towers supplies (per group): 2 pieces of paper, 10 paper clips, 1 pair of scissors Counters (or small place holders such as paper clips) 10 per pair of students.

3 3 Module 1 Vocabulary: Expression Equation Equivalent expressions/equations Inequality Less than/less than or equal to Greater than/greater than or equal to Literal equations Distribute Solve Simplify Context Slope y-intercept Coordinate plane Justify Evaluate Open/closed dot Units of measure Rate Absolute value Concepts Used In the Next Module: Graph linear equations & inequalities Solve linear equations & inequalities Define variables from a context Write equations from a context Determine if a given point is a solution to an equation, inequality, or systems of equations Solve systems of linear equations by graphing, substitution, & elimination Solve systems of linear inequalities by graphing Graph lines using technology (i.e. Graphing Calculators or Desmos)

4 Module 1 Equations and Inequalities Combining like terms, distributive property, and solving linear equations (A.REI.1, A.REI.3, A.SSE.1, N.Q.2) Warm Up: Group Cards Like Terms, Paper Towers and/or Silent Square Classroom Task: Checkerboard Borders A Develop Understanding Task and Building More Checkerboard Borders Ready, Set, Go Homework: Equations and Inequalities Interpreting expressions and using units to understand problems (A.SSE.1, N.Q.1) Warm Up: Serving Up Symbols A Develop Understanding Task Classroom Task: Examining Units A Solidify Understanding Task Ready, Set, Go Homework: Equations and Inequalities Explaining each step in the process of solving an equation (A.REI.1) Warm Up: Graph Linear Equations Classroom Task: Cafeteria Actions and Reactions A Develop Understanding Task Ready, Set, Go Homework: Equations and Inequalities Solving literal equations (A.REI.1, A.REI.3, A.CED.4) Warm Up: Pattern Representations Classroom Task: Solving Equations, Literally A Practice Understanding Task Ready, Set, Go Homework: Equations and Inequalities Writing inequalities to fit a context and Reasoning about inequalities and the properties of inequalities solving linear inequalities and representing the solution (A.REI.1, A.REI.3) Warm Up: Greater Than? A Solidify Understanding Task Classroom Task: Taking Sides A Practice Understanding Task Ready, Set, Go Homework: Equations and Inequalities Solving absolute value equations and inequalities (A.REI.3, A.REI.3.1) Warm Up: Solve Systems of Equations Classroom Task: Absolutely Sure A Practice Understanding Task Ready, Set, Go Homework: Equations and Inequalities 1.6 Module 1 Test Day Homework: Letter from Carlos and Clarita

5 Checkerboard Borders A Develop Understanding Task In preparation for back to school, the school administration has planned to replace the tile in the cafeteria. They would like to have a checkerboard pattern of tiles two rows wide as a border for the tables and serving carts. Below is an example of the border that the administration is thinking of using to surround a square 5 x 5 set of tiles. a. Find the number of colored tiles in the checkerboard border. Track your thinking and find a way of calculating the number of colored tiles in the border that is quick and efficient. Be prepared to share your strategy and justify your work.

6 b. The contractor that was hired to lay the tile in the cafeteria is trying to generalize a way to calculate the number of colored tiles needed for a checkerboard border surrounding a square of tiles with dimensions s x s. Find an expression for the number of colored border tiles needed for any s x s square center. 6

7 7 Building More Checkerboard Borders As the tile workers started to look more deeply into their work they found it necessary to develop a way to quickly calculate the number of colored border tiles for not just square arrangements but also for checkerboard borders to surround any L W rectangular tile center. Find an expression to calculate the number of colored tiles in the two row checkerboard border for any rectangle. Be prepared to share your strategy and justify your work. Create models to assist you in your work.

8 8 Name: Equations and Inequalities 1.1 Ready, Set, Go! Ready Topic: Create and solve equations in one variable. Use the pictures below to answer questions Each square represents one tile, how many total tiles are in Step 5? Step 6? 2. What might you do to determine the number of tiles in Step 25? Set Topic: Graph linear equations For the following problems, two points and a slope are given. Use the graph to plot these points, draw the line, and clearly label the slope on the graph. 3. and 4. and 5. and Slope: Slope: Slope:

9 9 For the following problems, two points are given. Use the graph to plot these points and find the slope. 6. and 7. and 8. and Slope: Slope: Slope: Go For problems 9 and 10, the y-intercept and the slope of a line are given. Graph the line on the coordinate axes, clearly labeling the slope and y-intercept The equations below are represented in the above graphs. Explain how the slope and y-intercept show up in both the graph and algebraic representations.

10 10 For problems 11 13, graph the following equations on the provided coordinate axes

11 Serving Up Symbols A Develop Understanding Task As you look around your school cafeteria, you may see many things that could be counted or measured. To increase the efficiency of the cafeteria, the cafeteria manager, Elvira, decided to take a close look at the management of the cafeteria and think about all the components that affect the way the cafeteria runs. To make it easy, she assigned symbols for each count or measurement that she wanted to consider, and made the following table: Symbol Meaning (description of what the symbol means in context) Units (what is counted or measured) S Number of students that buy lunch in the cafeteria each day students or students per day B Number of boys that buy lunch each day boys or boys per day G F T Number of girls that buy lunch each day Number of food servers in the cafeteria Total number of food items in one lunch (Each entrée, side dish, or beverage counts as 1 food item) Number of entrées in each lunch Number of side dishes in each lunch Number of beverages in each lunch Cost of each entrée Cost of each side dish Cost of each beverage L W i H P Number of lines in the cafeteria Number of food servers in each line Average number of food items that a server can serve each minute (Each entrée, side dish, or beverage counts as 1 food item) Number of hours each food server works each day Price per lunch 1. Using these symbols, what would the expression mean? 2. Using these symbols, what would the expression mean?

12 3. Determine if the given expressions makes sense. If it does, what does it mean in terms of the context? If it does not, explain why not. a. 12 b. c. 4. Elvira hopes to use the symbols in the chart to come up with some meaningful expressions that will allow her to analyze her cafeteria. Your job is to help her by writing as many expressions as you can and describe what they mean. Put your expressions in the following chart, adding lines if needed. Expression Description

13 2012 ww.flickr.com/photos.mahuskali Examining Units A Solidify Understanding Task (Note: This task refers to the same set of variables as used in Serving Up Symbols) Units in Addition and Subtraction 1. Why can you add and you can add, but you can t add? We measure real-world quantities in units like feet, gallons, students, and miles/hour (miles per hour). 2. State a rule about how you might use units to help you think about what types of quantities can be added. How would you use or modify your rule to include subtraction? Units in Multiplication, Scenario 1 3. Why can you multiply and you can multiply, but you can t multiply? Units in multiplication often involve rates like miles/gallon (miles per gallon), feet/second (feet per second), or students/table (students per table). An example from physics is. 4. What units might you use to measure and individually? What about the product? 5. An example from physics is, where is wavelength measured in meters and is frequency, or rate of time, measured per second, and c is the constant speed of light. Since the equation shows the multiplication, is equal to the speed of light constant, what are the units for c? 6. State a rule about how you might use units to help you think about what types of quantities can be multiplied. Units in Multiplication, Scenario 2 7. Let l represent the length of the cafeteria and w represent its width, in feet. What does the expression represent? What about? 8. Why can we add and multiply? What is it about these variables that allows them to be added or multiplied?

14 14 Units in Division, Scenario 1 What are the units for the dividend (what you are dividing up), the divisor (what you are dividing by), and the quotient (the result of the division) in the following expressions: Dividend: Divisor: Quotient: Dividend: Divisor: Quotient: 11. In chemistry, you will likely need to calculate the density of an object. To do this, you will divide the mass of an object (measured in grams), m, by the volume of the object (measured in milliliters). The formula is. What are the units for D? 12. State a rule about the units in division problems like those represented above. Units in Division, Scenario 2 What are the units for the dividend, the divisor, and the quotient in the following expressions: Dividend: Divisor: Quotient: Dividend: Divisor: Quotient: 15. State a rule about the units in division problems like those represented above.

15 15 Name: Equations and Inequalities 1.2 Ready, Set, Go! Ready Topic: Linear modeling 1. The local amusement park sells summer memberships for $50 each. Normal admission to the park costs $25; admission for members costs $15. a. If Darren wants to spend no more than $100 on trips to the amusement park this summer, how many visits can he make if he buys a membership with part of that money? b. How many visits can he make if he does not? c. If he increases his budget to $160, how many visits can he make as a member? d. How many can he make as a non-member? 2. Jae just took a math test with 20 questions, each worth an equal number of points. The test is worth 100 points total. a. Write an equation relating the number of questions Jae got right to the total score he will get on the test. b. If a score of 70 points earns a grade of C, how many questions would Jae need to get right to get a C on the test? c. If a score of 83 points earns a grade of B, how many questions would Jae need to get right to get a B on the test? d. Suppose Jae got a score of 60% and then was allowed to retake the test. On the retake, he got all the questions right that he got right the first time, and also got half the questions right that he got wrong the first time. What percent did Jae get right on the retake?

16 16 Set Topic: Using variables in context Variable B G L S M E Meaning the number of Boys in the classroom the number of Girls in the classroom the number of Pencils each Boy has (assume all boys have the same number of pencils) the number of Pencils each Girl has (assume all girls have the same number of pencils) the cost of a Lunch for each students (in cents) the cost of a Snack for each students (in cents) the amount of time each student spends in Math class per day (in minutes) the amount of time each student spends in English class per day (in minutes) the amount of time each student spends on Homework for Math per day (in minutes) the amount of time each student spends on Homework for English per day (in minutes) 3. Use the variables and their meanings from above to describe what each algebraic expression means (if anything). a. b. c. d. 4. Write an algebraic expression for each phrase a. The total number of pencils for the students in the class b. The cost of lunch for the whole class c. The total amount of time students in the class spend on English each day, both in class and on homework

17 17 Go Topic: Evaluating expressions Complete each of the following: 5. Evaluate at. 6. Find the value of when. 7. Get the numeric value of for. 8. Evaluate if. 9. What is when? 10. Find with. 11. Compare the results. a. Substitute into. b. Substitute into. c. Comment on what happens in parts a and b and explain why.

18 Warm Up Graphing lines in slope-intercept form Graph the following equations on the provided coordinate grids Solve each equation, justify each step you use. For number 4, the first step is shown for you. You must finish it Justification Subtract x from both sides to get the x s on one side. Justification

19 Cafeteria Actions and Reactions A Develop Understanding Task Elvira, the cafeteria manager, has just received a shipment of new trays with the school logo prominently displayed in the middle of the tray. After unloading 4 cartons of trays in the pizza line, she realizes that students are arriving for lunch and she will have to wait until lunch is over before unloading the remaining cartons. The new trays are very popular and in just a couple of minutes 24 students have passed through the pizza line and are showing off the school logo on the trays. At this time, Elvira decides to divide the remaining trays in the pizza line into 3 equal groups so she can also place some in the salad line and the sandwich line, hoping to attract students to the other lines. After doing so, she realizes that each of the three serving lines has only 12 of the new trays. That s not many trays for each line. I wonder how many trays there were in each of the cartons I unloaded. 1. Can you help the cafeteria manager answer her question using the data in the story about each of the actions she took? Explain how you arrive at your solution.

20 20 Elvira is interested in collecting data about how many students use each of the tables during each lunch period. She has recorded some data on Post It Notes to analyze later. Here are the notes she has recorded: Some students are sitting at the front table. (I got distracted by an incident in the back of the lunchroom, and forgot to record how many students.) Each of the students at the front table has been joined by a friend, doubling the number of students at the table. Four more students have just taken seats with the students at the front table. The students at the front table separated into three equal sized groups. One group remained and all the other students left. As the lunch period ends, there are still 12 students seated at the front table. 2. Elvira is wondering how many students were sitting at the front table when she wrote her first note. Unfortunately, she is not sure what order the middle three Post It Notes were recorded in since they got stuck together in random order. She is wondering if it matters. Does it matter which order the notes were recorded in? Determine how many students were originally sitting at the front table based on the sequence of notes that appears above. 3. Rearrange the middle three notes in a different order and determine what the new order implies about the number of students seated at the front table at the beginning. If the result is the same, explain or show why. If a different order gives a new result, is there a specific order that gives the highest total?

21 4. Here are three different equations that could be written based on a particular sequence of notes. Examine each equation, and then list the order of the five notes that is represented by each equation. Find the solution for each equation. 21 ( )

22 22 Name: Equations and Inequalities 1.3 Ready, Set, Go! Ready Topic: Solutions to an equation Graph the following equations using the coordinate graph, then say if the given point is a solution to the equation Point: Yes/No Point: Yes/No Point Yes/No Point Yes/No Point Yes/No Point Yes/No

23 23 Set Topic: Equivalent Equations 7. The solution to an equation is. The equation has parentheses on at least one side of the equation and has variables on both sides of the equation. What could the equation be? 8. Create a two-step equation with the given solution. Draw a model to represent your expanded equations. a. b. c. 9. Without solving, determine if the left and right sides of each equations are equivalent. Explain your reasoning. a. b. 10. Without solving, determine if these two equations have the same solution. and 11. Which of the following expressions are equivalent?

24 Solving equations can be similar to finding the unkown weights on a balance scale like the one shown below. Each rectangle represents an unknown amount. Each circle represents a weight of one unit. Some of these are grouped together as shown. Question 12 is drawn out for you. Your task is to fill in the equations that are represented in each step of the diagram. The first few are begun for you Translate the diagrams into the four algebraic equations:

25 25 Go Topic: Checking solutions to equations Check whether the given number is a solution to the corresponding equation. 14. ; 15. ; 16. ; 17. ; Topic: Solving multi-step linear equations Solve each equation, justify each step you use Justification Justification

26 Warm Up Pattern Representations Topic: Create a rule based on a pattern Use the pictures below to answer questions 1 3. Step 1 Step 2 Step 3 1. Each square represents one tile, how many total tiles are in Step 5? Step 6? 2. What might you do to determine the number of tiles in Step 25? 3. Complete the table and plot the points on the grid. Step # Number of Tiles

27 Solving Equations, Literally A Practice Understanding Task Solve each of the following equations for x: Solve the equation for the variable m. 10. Solve the equation for the variable R.

28 11. Describe each step you use to solve the following equations for x. Your description should include statements about how you know what to do next. For example, you might write, First I, because Justification

29 29 Name: Equations and Inequalities 1.4 Ready, Set, Go! Ready Topic: Inequalities Use the inequality to complete each row in the table. Apply each operation to the original inequality Result Is the inequality true or false? 1. Add 4 to both sides 2. Add to both sides 3. Subtract 10 from both sides 4. Multiply both sides by 4 5. Divide both sides by 2 6. Multiply both sides by 7. Divide both sides by 8. Which operations in the table create a false inequality? What do they have in common? 9. In general, what operations, when performed on an inequality, reverse the inequality? 10. Solve the inequality and graph the solution on a number line. a. b.

30 30 Set Topic: Solve literal equations Solve for the indicated variable. 11. Solve the following equation to isolate F: 12. For, rewrite the formula to isolate the variable h. 13. The area formula of a regular polygon is The variable a represents the apothem and P represents the perimeter of the polygon. Rewrite the equation to highlight the value of the perimeter, P. 14. The equation is the equation of a line. Isolate the variable m. 15. The equation is the equation of a line. Isolate the variable x. 16. is the standard form of a line. Isolate the equation for x.

31 31 Rearrange the following equations to solve for y (slope-intercept form) Go Topic: Solve the systems of linear equations Solve linear equations and pairs of simultaneous linear equations (simple, with a graph only) by graphing both lines and finding where they intersect. Justify the solution numerically by checking your solutions in both equations. 23. and 24. and and

32 Warm Up: Greater Than? A Solidify Understanding Task For each situation you are given a mathematical statement and two expressions beneath it. 1. Decide which of the two expressions is greater, if the expressions are equal, or if the relationship cannot be determined from the statement. 2. Write an equation or inequality that shows your answer. 3. Explain why your answer is correct. Watch out this gets tricky! Example: Statement: Which is greater? or Answer: because if,, and. Try it yourself: 1. Statement: Which is greater? 1 or x 2. Statement: Which is greater? 5 or x 3. Statement: Which is greater? x or 2 4. Statement: Which is greater? or 5. Statement: Given the number line at right Which is greater? 1 or

33 Taking Sides A Practice Understanding Task Joaquin and Serena work together productively in their math class. They both contribute their thinking and when they disagree, they both give their reasons and decide together who is right. In their math class right now, they are working on inequalities. Recently they had a discussion that went something like this: Joaquin: The problem says that 6 less than a number is greater than 4. I think that we should just follow the words and write. Serena: I don t think that works because if x is 20 and you do 6 less than that you get. I think we should write. Joaquin: Oh, you re right. Then it makes sense that the solution will be means we can choose any number greater than 10., which The situations below are a few more of the disagreements and questions that Joaquin and Serena have. Your job is to decide how to answer their questions, decide who is right, and give a mathematical explanation of your reasoning. 1. Joaquin is thinking hard about equations and inequalities and comes up with this idea: If, then. So, if, then. Is he right? Explain why or why not. 2. Joaquin s question in #1 made Serena think about other similarities and differences in equations and inequalities. Serena wonders about the equation and the inequality. Explain to Serena ways that solving these two problems are alike and ways that they are different. How are the solutions to the problems alike and different?

34 3. Joaquin solved by adding 15 to each side of the inequality. Serena said that he was wrong. Who do you think is right and why? 34 Joaquin s solution was. He checked his work by substituting 150 for q in the original inequality. Does this prove that Joaquin is right? Explain why or why not. Joaquin is still skeptical and believes that he is right. Find a number that satisfies his solution but does not satisfy the original inequality. 4. Serena is checking her work with Joaquin and finds that they disagree on a problem. Here s what Serena wrote: Is she right? Explain why or why not? 5. Joaquin and Serena are having trouble solving. Explain how they should solve the inequality, showing all the necessary steps and identifying the properties you would use.

35 6. Joaquin and Serena know that some equations are true for any value of the variable and some equations are never true, no matter what value is chosen for the variable. They are wondering about inequalities. What could you tell them about the following inequalities? Do they have solutions? What are they? How would you graph their solutions on a number line? a. b. c The partners are given the literal inequality to solve for x. Joaquin says that he will solve it just like an equation. Serena says that he needs to be careful because if a is a negative number, the solution will be different. What do you say? What are the solutions for the inequality?

36 36 Name: Equations and Inequalities 1.5 Ready, Set, Go! Ready Topic: Solve absolute value equations Find values of x that make each equation true Set Topic: Solve inequalities Solve each inequality and graph the solution on a number line

37 Solve each multi-step inequality

38 For each situation you are given a mathematical statement and two expressions beneath it. a. Decide which of the two expressions is greater, if the expressions are equal, or if the relationship cannot be determined from the statement. b. Write an equation or inequality that shows your answer. c. Explain why your answer is correct. 16. Statement: Which is greater? n or Statement: Which is greater? 5x or 4x 18. Statement Which is greater? or 19. Statement: Which is greater? or Go Topic: Systems of equations Write two equations to represent the following situation. 20. A tortoise and hare decide to race 30 feet. The hare, being much faster, decides to give the tortoise a 20 foot head start. The tortoise runs at 0.5 feet/sec and the hare runs at 5.5 feet per second. How long until the hare catches the tortoise?

39 Warm Up Solve systems of equations When sketching a graph function, it is important that we see important points. For linear functions, we want a window that shows important information related to the story. Often, this means including both the -and -intercepts. Note: is called interval notation and is the same as saying the values are greater than or equal to and less than or equal to 10. In inequality notation, this looks like. Example: Window: by x-scale: 1 and y-scale: 1 NOT a good window Window: by x-scale: 5 and y-scale: 5 Good window For the following equations, state a window and scale for each axis that would be satisfactory for the given equation. Then sketch a graph in the boxes provided Use a graphing utility to solve. Be sure to write down your equations and a sketch of the graph from the utility. A tortoise and hare decide to race 30 feet. The hare, being much faster, decides to give the tortoise a 20 foot head start. The tortoise runs at 0.5 feet/sec and the hare runs at 5.5 feet per second. How long until the hare catches the tortoise?

40 Absolutely Sure A Practice Understanding Task You and your partner are to solve the following absolute value equations and inequalities using the number line provided on the resource page. For each question, take turns placing a counter (or some type of place holder) on a value that is a solution. Once all of the solutions have been accounted for, sketch a number line below each problem and shade the solutions. Remember: Absolute value of x, written, means the distance a number x is from 0. Part Part Part

41 41 Part Will all absolute value equations have two solutions? Why or why not? 20. What do you notice about the pattern of the absolute value with? What would be different if we used < in Part 2? 21. What do you notice about the pattern of the absolute value with? What would be different if we used > in Part 3?

42 Resource Page

43 43 Name: Equations and Inequalities 1.6 Ready, Set, Go! Ready Topic: Solve and justify one variable inequalities Solve each inequality, justifying each step you use. 1. Justification 2. Justification 3. Justification 4. Justification Set Topic: Solving absolute value equations and inequalities Solve each inequality and graph the solution on the number line. 5.

44 44 6. Topic: Solving multi-step inequalities Topic: Solve literal equations 11. The equation is the equation of a line. Isolate the variable b. 12. For, rewrite the formula to isolate the variable h. 13. Solve the following equation to isolate B: 14. The equation for the circumference c of a circle with radius r is. Solve the equation for the radius r.

45 45 Solve the following equations for the unknown variable Calculate the slope. Be sure to simplify your answers. 17. and 18. and 19. and 20. and Write the equation of the line in slope intercept form by solving for y. Then graph each line

46 46 For #23-26, solve each inequality. Graph the solutions on the number line AND state 3 numbers in the solution set. Show all your work! numbers in the solution set: numbers in the solution set: numbers in the solution set: numbers in the solution set:

47 47 Go Topic: Solve systems of equations Graph the following pairs of linear equations on the same grid. Find the point of intersection. Justify the solution numerically. 27. { 28. {

48 48 Dear Integrated Math 1 Student, Carlos and I are about to start a business in order to make money for our summer vacation. Our aunt has been complaining recently about how hard it is to find quality and affordable care for her pets. This will be the perfect opportunity to help our aunt out and to support ourselves. Our parents have given us permission to start our own pet sitting business in our backyard, which we ll call Carlos and Clarita s Quality Creature Care. We have a large, unused storage shed and a spacious fenced backyard where the pets can play. Before we can dive into this business, we need your help in brainstorming things that we need to take into consideration when starting and running Carlos and Clarita s Quality Creature Care in order to maximize our profits. Thanks for your help! Sincerely, Carlos and Clarita Martinez