Solving Problems with Order of Operations
Home Link 4-2
Expressions and Patterns
Home Link 4-3
Representing Unknown Quantities with Algebraic Expressions
Home Link 4-4
Exploring Equations
Home Link 4-5
The Distributive Property and Equivalent Expressions
Home Link 4-6
Applying Properties of Arithmetic
Home Link 4-7
The Banquet Table
Home Link 4-8
Introduction to Inequalities
Home Link 4-9
Finding and Graphing Solution Sets of Inequalities
Home Link 4-10
Inequalities to Represent Real-World Situations
Home Link 4-11
Absolute Value as Distance
Home Link 4-12
Absolute Value
Home Link 4-13
Mean Absolute Deviation
Home Link 4-14
Unit 4 Progress Check
Home Link 4-15
Everyday Mathematics for Parents: What You Need to Know to Help Your Child Succeed
The University of Chicago School Mathematics Project
University of Chicago Press
Learn more >>
Help with algorithms.
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Learn more about the EM curriculum and how to assist your child.
Working with Trigonometric Ratios
CCSS Standards
This warm-up uses a problem students didn’t know how to approach at the beginning of the unit. At this point they can try again using the right triangle table. Then when students learn to look up trigonometric ratios in the calculator, that procedure will be solidly grounded in conceptual understanding.
Estimate the value of \(z\) .
For access, consult one of our IM Certified Partners .
Point out to students this is the triangle they didn’t know how to approach at the beginning of the unit. Now they have a strategy for finding side lengths of right triangles for acute angles included in the right triangle table. The purpose of this lesson is to teach a method that works for any angle, not just the ones in the table.
Routines and Materials
Instructional Routines
The purpose of this activity is to connect the work students have done in previous lessons and the warm-up to trigonometric ratios. Students learn the names of the trigonometric ratios and how to look them up in the calculator. To continue solidifying their conceptual understanding, students compare the calculator’s value to their work with the right triangle table.
Tell students, “The right triangle table is useful, but what if the angle is not a multiple of 10 degrees? There must be a way to move from estimating to calculating. The name of the column in the right triangle table, 'adjacent leg divided by the hypotenuse,' is long, so mathematicians call this ratio cosine . The column called 'opposite leg divided by the hypotenuse' is named sine . The column called 'opposite leg divided by the adjacent leg' is named tangent . These three names describe trigonometric ratios . Label the columns of the right triangle table with the corresponding trigonometric function names.”
“The word trigonometric comes from the ancient roots of trigon (think about what pentagon or hexagon mean) and metric (like measure). So the word trigonometric comes from ancient words meaning triangle measurement.”
“Cosine, sine, and tangent work like functions where the input is the measure of an angle and the output is a ratio. Scientific calculators can display the ratio for any angle, even ones not included in the table. First, identify that we have an angle measure and the length of the adjacent leg. Then identify that we are looking for the length of the hypotenuse. The ratio of adjacent leg divided by hypotenuse is called cosine. So we can write \(\cos(50)=\frac{3}{z}\) .” Demonstrate how to use the technology available to solve this type of problem. Typing \(\cos(50)\) into the calculator and then substituting gives the new equation \(0.643=\frac{3}{z}\) . (Note: if students don’t get 0.643, check the mode of the calculator. There is no need to discuss the meaning of radians now, but students will see radians later in this course.)
Invite students to share their new value for \(z\) . (Solving the equation we wrote in the launch gives \(z=4.7\) units.)
Ask students:
Inform students it is also okay to leave answers in the form \(z=\frac{3}{\cos(50)}\) which would be exact.
Ask students to add these definitions to their reference charts as you add them to the class reference chart:
The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse.
\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\)
The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse.
\(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)
The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg.
\(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\)
Building On
Required Materials
Students apply their new knowledge of trigonometric ratios to solve these problems. Since these problems ask for multiple side and angle measures, there is an increased opportunity for creativity in solving.
Monitor for students who solve multiple trigonometric equations versus those that apply the Pythagorean Theorem.
Arrange students in groups of 2. Ask students to compare their strategy with their partner’s and decide if they are both correct, even if they are different.
Students need not complete all the problems before the discussion.
Solve for \(x\) .
Solve for \(y\) .
Find all the missing sides and angle measures.
The measure of angle \(X\) is 90 degrees and angle \(Y\) is 12 degrees. Side \(XZ\) has length 2 cm.
The measure of angle \(K\) is 90 degrees and angle \(L\) is 71 degrees. Side \(LM\) has length 20 cm.
Complete the table.
angle | cosine | sine | tangent |
---|---|---|---|
\(80^\circ\) | |||
\(85^\circ\) | |||
\(89^\circ\) |
Based on this information, what do you think are the cosine, sine, and tangent of 90 degrees? Explain or show your reasoning.
If students struggle to get started, prompt them to set up ratios to find the missing sides. If students struggle to set up ratios, prompt them to identify what is known and what they are looking to find by annotating the diagram.
Invite students to contrast solving multiple trigonometric equations versus applying the Pythagorean Theorem to determine the final side length. Is one method easier? More accurate? (The Pythagorean Theorem is more familiar so I prefer it. Both methods are accurate so long as you don't round too much.)
Call attention to the triangle with no marked right angle. Ask students, “How do you know you can use trigonometric functions with this triangle?” (By the Triangle Angle Sum Theorem the third angle must be 90 degrees.)
Display this triangle and give students quiet work time to write as many equations as they can to show the relationships in triangle \(END\) .
Invite students to share one equation at a time until you have recorded two of each trigonometric ratio ( cosine, sine, and tangent ):
\(\cos(35)=\frac{d}{n}\)
\(\sin(35)=\frac{e}{n}\)
\(\tan(35)=\frac{e}{d}\)
\(\cos(55)=\frac{e}{n}\)
\(\sin(55)=\frac{d}{n}\)
\(\tan(55)=\frac{d}{e}\)
Tell students the value of \(d\) is 6 units. First, ask them to consider which equations would be helpful in solving for the other sides. Then give them time to do the calculations ( \(e=4.2 \) units, \(n=7.3\) units). If students get slightly different answers, take the opportunity to discuss precision and rounding error.
Student lesson summary.
We have a column in the right triangle table for "adjacent leg \(\div\) hypotenuse." We use this ratio so frequently it has a name. It is called the cosine of the angle. We write \(\cos(25)\) to say the cosine of 25 degrees. A scientific calculator can display the cosine of any angle. This means we can more precisely calculate unknown side lengths rather than estimating using the table. The right triangle table is sometimes called a trigonometry table since cosine, sine , and tangent are trigonometric ratios . Here is what the table looks like with the ratios labeled with their special names:
cosine | sine | tangent | |
---|---|---|---|
angle | adjacent leg \(\div\) hypotenuse | opposite leg \(\div\) hypotenuse | opposite leg \(\div\) adjacent leg |
\(25^\circ\) | \(\cos(25)=0.906\) | \(\sin(25)=0.423\) | \(\tan(25)=0.466\) |
If the length \(b\) is 7, we can find \(c\) by solving the equation \(\cos(25)=\frac{7}{c}\) . So \(c\) is about 7.7 units. To solve for \(a\) we have 3 choices: the Pythagorean Theorem, sine, and tangent. Let’s use tangent by solving the equation \(\tan(25)=\frac{a}{7}\) . So \(a\) is about 3.3 units. We can check our answers using the Pythagorean Theorem. It should be true that \(3.3^2+7^2=7.7^2\) . The two expressions are almost equal, which makes sense because we expect some error due to rounding.
Introduction to Linear Functions
LESSON/HOMEWORK
LECCIÓN/TAREA
LESSON VIDEO
EDITABLE LESSON
EDITABLE KEY
SMART NOTEBOOK
More Work Graphing Linear Functions (Lines)
Writing the Equation of a Line
Working with Linear Functions in Table Form
Modeling with Linear Functions
More Modeling with Linear Functions
Equations of Horizontal and Vertical Lines
Piecewise Linear Functions
Step Functions
Absolute Value Functions
The Truth About Graphs
Linear Inequalities with Two Variables
Introduction to Sequences
Arithmetic Sequences
Unit Review
Unit 4 Review
UNIT REVIEW
REPASO DE LA UNIDAD
EDITABLE REVIEW
Unit 4 Assessment – Form A
EDITABLE ASSESSMENT
Unit 4 Assessment – Form B
Unit 4 Exit Tickets
Unit 4 Mid-Unit Quiz – Form A
U04.AO.01 – Modeling the Boundaries of a Shed
PDF DOCUMENT - SPANISH
EDITABLE RESOURCE
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Go Math Grade 6 Answer Key: In Today’s World learning has become demanding than ever before. Finding a reliable source has become a tedious task for anyone out there who wants to upgrade their skills. Sharpen your Math Skills taking the help of 6th Grade Go Math Middle School Solutions Key en route to your math Journey. Middle School Go Math Grade 6 Solutions Key includes Worked Out Solutions for all the Problems in Go Math Textbooks.
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All the solutions of Middle School Go Math Books for Grade 6 are prepared by subject experts. Go Math Books prevailing for 6th Standard are prepared to meet both the content and intent of Middle School. You can witness the Mathematical Concepts explained in a concise manner making it easier for you to have a good grip on the subject. HMH Go Math Grade 6 Answer Key includes the solved examples and practice questions to strengthen your Mathematical Concepts.
Avail Grade 6 Solutions provided over here and understand the concepts in a better way. Identify the Knowledge Gap and allot time to the areas you feel difficult. Detailed description provided in the Go Math Grade 6th Solutions Key reflects more of the topics in your Middle School Textbooks. You can use them during your Homework or while preparing for Tests. Tap on the respective chapter you wish to practice and clarify all your concerns at one go.
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Eureka Math Grade 4 Module 7 Lesson 2 Answer Key; Eureka Math Grade 4 Module 7 Lesson 3 Answer Key; Eureka Math Grade 4 Module 7 Lesson 4 Answer Key; Eureka Math Grade 4 Module 7 Lesson 5 Answer Key; Eureka Math 4th Grade Module 7 Topic B Problem Solving with Measurement. Eureka Math Grade 4 Module 7 Lesson 6 Answer Key
Lesson 6: Solve problems involving mixed units of capacity. Lesson 6 Homework 4• 7 3. The capacity of a bucket is 5 gallons. Right now, it contains 3 gallons 2 quarts of liquid. How much more liquid can the bucket hold? 4. Grace and Joyce follow the recipe in the table to make a homemade bubble solution. a. How much solution does the recipe ...
Module 7: Exploring Measurement with Multiplication Lesson 2 Answer Key 4• 7 Exit Ticket 1. 4, 8, 16 2. No; answers will vary. Homework 1. 48 cups 6. Answers will vary. 2.
It's Homework Time! Help for fourth graders with Eureka Math Module 7 Lesson 6.
Lesson 2 Homework 4• 6 Lesson 2: Use metric measurement and area models to represent tenths as fractions greater than 1 and decimal numbers. Name Date 1. For each length given below, draw a line segment to match. Express each measurement as an equivalent mixed number. a. 2.6 cm b. 3.5 cm c. 1.7 cm d. 4.3 cm e. 2.2 cm 2.
Eureka Math Grade 4 Module 4 Lesson 7 Practice Sheet Answer Key. Figure 1. Answer: The angle WYX would be 60° or above but less 90° and the angle is called an acute angle. Explanation: Here, the angle <WYX would be an acute angle. As In the above image, we can see that the angle is less than 90°.
Lesson Check (CC.5.NBT.2, CC.5.NBT.7) 1. Sue buys material to make a costume. She buys 1.75 yards of red material. She buys 1.2 times as many yards of blue material. How many yards of blue material does Sue buy? 2.1 yards 2.95 yards C) 5.25 yards @ 21 yards 2. Last week Juan worked 20.5 hours.
Answer Key Eureka Math速 Grade 4 Module 4. Special thanks go to the Gordon A. Cain Center and to the Department of Mathematics at Louisiana State University for their support in the development of. Eureka Math. Answer Key. Eureka Math速 Grade 4 Module 4. TEKS EDITION. Module 4:Angle Measure and Plane Figures269.
Answer Key Lesson 4.7 Practice Level C 1. x 5 9, y 5 11 2. x 5 6, y 5 13 3. x 5 3.5, y 5 9 4. x 5 12, y 5 5 5. x 5 6, y 5 7 6. x 5 3, y 5 9.5 7. x 5 20.5, y 5 6 8. x 5 8, y 5 3 9. x 5 7, cannot determine y; could findy if it was given that 9y 2 10 is equal to 5y 2 8. 10. 98 in. 11. 72.5 m 12. 149.25 ft
4•Lesson 1 Answer Key Lesson 1 Problem Set 1. a. - f. Figure drawn accurately g. Answers will vary. 2. a. - g. Figure drawn accurately h. Answers will vary. 3. a. Points labeled; labels will vary . b. Answers will vary. Exit Ticket 1. Words connect to corresponding picture s 2. Answers will vary. Homework 1. a. - f. Figure drawn ...
Lesson 8: Solve problems involving mixed units of weight. Lesson 8 Homework 4• 7 Notebook 11 oz Laptop 5 lb 12 oz Textbook 3 lb 8 oz Supply Case 1 lb Backpack (empty) 2 lb 14 oz Binder 2 lb 5 oz 4. Use the information in the chart about Melissa's school supplies to answer the following questions: a. On Wednesdays, Melissa packs only
Answer Key Eureka Math® Grade 2 Module 4 TEKS EDITION Special thanks go to the Gordon A. Cain Center and to the Department of Mathematics at Louisiana State University for their support in the development of
Eureka Math Grade 4 Module 6 Lesson 4 Exit Ticket Answer Key. Question 1. Shade in the amount shown. Then, write the equivalent decimal. 0.6 m. 6/10 = 0.6. Question 2. Draw a number bond, pulling out the tenths from the hundredths. Write the total as the equivalent decimal.
Lesson 7.8 COMMON CORE STANDARDS CC.5.NF.5a, CC.5.NF.5b Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 3 equal to 3. 5. less than 15 x lg will be Think: 1 X is lg. Since is less than 1, X 15 will be less than 15. greater than 3 x 33 will be greater than 12 x 23 will be 2. -x '234 4. 9 X 14
Selected Answers. 4-6. ... Applying Properties of Arithmetic. Home Link 4-7 English Español Selected Answers. 4-8. The Banquet Table. Home Link 4-8 English Español Selected Answers. 4-9. Introduction to Inequalities ... access resources to help your child with homework or brush up on your math skills.
93 4 7 about 15 M 3. 125 4 6 about 12 I 4. 74 4 5 about 20 T 5. 275 4 4 about 70 E 6. 35 4 3 about 40 A 7. 481 4 9 about 18 T 8. 127 4 7 about 13 S Estimating Quotients Match each quotient with its best estimate. Then write the letter of the estimate on the appropriate blank to answer the question below. Lesson 4.1 Enrich
View Student Lesson. 6.1: This Time with Strategies (5 minutes) CCSS Standards. Addressing. HSG-SRT.C.6; Warm-up. This warm-up uses a problem students didn't know how to approach at the beginning of the unit. At this point they can try again using the right triangle table. ... "How does this answer compare to using the table?" (It's the ...
Download Lesson 6 Homework 4.7 Answer Key: FileName. Speed. Downloads. Lesson 6 Homework 4.7 Answer Key | updated. 5411 kb/s. 6580. Lesson 6 Homework 4.7 Answer Key | NEW. 1302 kb/s. ... Gizmo comes with an answer key. Each lesson includes a Student Exploration Sheet, an Exploration Sheet Answer Key, a Teacher Guide, a Vocabulary Sheet and ...
Eureka Math Grade 4 Module 4 Lesson 9 Problem Set Answer Key. Question 1. Complete the table. Refer to the below table. In pattern block A, the total number that fit around one vertex is 4 and the interior angle measurement is 360° ÷ 4 which is 90° and the sum of the angles around the vertex is 90°+90°+90°+90°= 360°.
LESSON/HOMEWORK. LECCIÓN/TAREA. LESSON VIDEO. ANSWER KEY. EDITABLE LESSON. ... is a Single User License, which means it gives one person - you — the right to access the membership content (Answer Keys, editable lesson files, pdfs, etc.) but is not meant to be shared. ... please credit us as follows on all assignment and answer key pages ...
Eureka Math Grade 4 Module 7 Lesson 12 Problem Set Answer Key. Question 1. Draw a tape diagram to show 1 yard divided into 3 equal parts. Question 2. Draw a tape diagram to show 2 yards = 8 feet. Question 3. Draw a tape diagram to show gallon = 3 quarts. Question 4. Draw a tape diagram to show 3 gallons = 15 quarts.
Find Math, English language arts (ELA) resources to practice & prepare lesson plans online with pdf, answer key, videos, apps, and worksheets for grades 3-8 on Lumos Learning. Login here Textbook Resources. Search: Grade 3; Grade 4; Grade 5; Grade 6; Grade 7 ... 6: 2014: Grade 6 ELL With Answers: Lumos Learning: 6: 2014: SBAC Test Prep: 6th ...
Grade 6 HMH Go Math - Answer Keys. Chapter 1: Divide Multi-Digit Numbers. Chapter 2: Fractions and Decimals. Chapter 3: Understand Positive and Negative Numbers. Chapter 4: Model Ratios. Chapter 5: Model Percents. Chapter 6: Convert Units of Length. Chapter 7: Exponents. Chapter 8: Solutions of Equations.