Name Period Date 7-CORE3.1 Geometric Figures Measure and draw angles using a protractor. Review facts about interior angles of triangles and quadrilaterals. Find missing angle measures in geometric diagrams. 7-CORE3.2 Angle Explorations Understand facts about supplementary, complementary, vertical, and adjacent angles. Use facts about angles to solve problems. Write and solve equations involving angle measures. 7-CORE3.3 Drawing Geometric Figures Draw figures freehand, with a ruler and protractor, and using technology. Construct triangles given side lengths and angle measures. Recognize when conditions determine a unique triangle, more than one triangle, or no triangle. 7-CORE3.4 Modeling With Expressions Describe geometric patterns numerically, pictorially, symbolically, and verbally. Interpret expressions in terms of their geometric context. 7-CORE3 STUDENT PACKET GRADE 7: MATHEMATICS COMMON CORE SUPPLEMENT ANGLES, DRAWINGS, AND ALGEBRAIC CONNECTIONS 7-CORE3.5 Vocabulary, Skill Builders, and Review 26 1 7 14 21 7-CORE3 SP
WORD BANK Word or Phrase Definition or Explanation Example or Picture adjacent angles angle complementary angles expression quadrilateral supplementary angles triangle vertical angles 7-CORE3 SP0
3.1 Geometric Figures Ready (Summary) We will use a protractor to measure and draw angles. We will review facts about triangles and quadrilaterals. GEOMETRIC FIGURES Go (Warmup) Set (Goals) Measure and draw angles using a protractor. Review facts about interior angles of triangles and quadrilaterals. Find missing angle measures in geometric diagrams. 1. An acute angle is an angle that measures 90. more than, less than, or exactly 2. A right angle is an angle that measures 90. more than, less than, or exactly 3. An obtuse angle is an angle that measures 90, but less than 180. 4. A straight angle is an angle that measures. Refer to the angle at the right for problems 5-6: more than, less than, or exactly 5. Which one of the labeled points represents the vertex of the angle? 6. Circle all names that can correctly be used to name this angle. ABC BCA BAC ACB C A Refer to the diagram at the right for problems 7-8. 7. A triangle has sides and angles. Name three different triangles in the diagram. 8. A quadrilateral has sides and angles. Name two different quadrilaterals in the diagram. C A B K M N P Q R 7-CORE3 SP1
3.1 Geometric Figures MEASURING ANGLES WITH A PROTRACTOR First state whether each angle appears to be acute, right, obtuse, or straight. Then use a protractor to find and record the measure. 1. 2. 3. 4. 5. 6. 7. 8. 9. Lee measured the angle in problem 8 and got 60 o. Write a sentence explaining why this is incorrect and how Lee can avoid this mistake. 7-CORE3 SP2
3.1 Geometric Figures DRAWING ANGLES WITH A PROTRACTOR First state whether each measure represents an acute, right, straight, or obtuse angle. Then use a protractor to draw the angle. One ray of the angle is given. 1. 60 2. 100 3. 30 4. 180 5. 90 6. 145 7. 10 8. 90 7-CORE3 SP3
3.1 Geometric Figures ANGLES PRACTICE First state whether each angle appears to be acute, right, or obtuse. Then use a protractor to find and record the measure. 1. 2. 3. 4. First state whether each measure represents an acute, right, or obtuse angle. Then use a protractor to draw the angle. One ray of the angle is given. 5. 80 7. 150 6. 45 8. 15 7-CORE3 SP4
3.1 Geometric Figures TEAR IT UP 1. Use a protractor to measure each angle in the triangle. m a = m b = m c = The sum of the measures of these three angles is. 2. Use a protractor to measure each angle in the parallelogram. m e = m f = m g = m h = The sum of the measures of these four angles is. Your teacher will give you some directions about tearing angles from triangles and quadrilaterals before doing problems 3 and 4. 3. The sum of the measures of the interior angles in a triangle is because Does this agree with your results in problem 1 above? Explain. 4. The sum of the measures of the interior angles in a quadrilateral is because Does this agree with your results in problem 2 above? Explain. Sketch your diagram here. Sketch your diagram here. h e a b g c f 7-CORE3 SP5
3.1 Geometric Figures FIND THE MISSING ANGLE MEASURES Use facts about angles, triangles, and quadrilaterals to write the missing angle measures. Do not use a protractor. Diagrams may not be to scale. 1. m r = 2. m p = 3. m n = 4. m z = 5. m y = 6. m x = 50 r 45 z 7. m w = 8. m v = 9. m u = w v 85 y p 60 60 x 95 85 55 62 65 u n 48 115 80 7-CORE3 SP6
3.2 Angle Explorations Ready (Summary) We will use patterns to learn facts about angles. We will use these facts to write equations and solve for unknown angle measures in diagrams. A D C Use the diagrams above. ANGLE EXPLORATIONS E B Go (Warmup) Set (Goals) Understand facts about supplementary, complementary, vertical, and adjacent angles. Use facts about angles to solve problems. Write and solve equations involving angle measures. 1. Name two right angles. 2. Name two acute angles. 3. Name two straight angles. 4. Name two obtuse angles. G F K H 7-CORE3 SP7
3.2 Angle Explorations A D SPECIAL ANGLES Use the two diagrams above and the definitions below to name the angle pairs. 1. Adjacent angles are angles that share a common vertex and a common side, and that lie on opposite sides of the common side. and and 3. Complementary angles are two angles with measures whose sum is 90 o. Name two pairs of complementary angles. and and C B E 2. Vertical angles are the opposite angles formed by two lines that intersect at a point. and and 4. Supplementary angles are two angles with measures whose sum is 180 o. Name two pairs of supplementary angles. and and 5. If complementary angles are congruent, what is the measure of each angle? 6. If supplementary angles are congruent, what is the measure of each angle? 7. Vanessa thinks that ACD and BCE cannot be vertical angles because they are in a horizontal orientation. Why is Vanessa incorrect? G F K H 7-CORE3 SP8
3.2 Angle Explorations A TRIANGLE PATTERN 1. The sum of the angle measures in the interior of a triangle is. 2. Consider the large rectangle below, which contains some congruent right triangles. Using three different colors, lightly shade in a, g, and h. a g b c e d f 3. For each set of angles that appear to be congruent, lightly shade them in with the same colors. Which angles appear to have the same measures as the following? a g h 4. Using a straightedge, duplicate the triangle pattern from the top rectangle in the larger rectangle below it. Label angles r, s, t, u, v, and w as instructed by your teacher. 5. Lightly shade these new angles to match their congruent partners above. Which of these angles appear to have the same measures as the following? a g h 6. Label the right angles in the diagram. h k Suppose m k = 55. Find the following angle measures. Support each answer with an explanation or calculation. 7. m b = 8. m h = 9. m u = 10. m w = y n p 7-CORE3 SP9
3.2 Angle Explorations A TRIANGLE PATTERN (Continued) Suppose m k = 55. Find the following angle measures. Support each answer with an explanation or calculation. 11. m e + m f + m p = 12. m h + m k + m y = 13. m a + m b + m c + m k + m h + m g = 14. Name a pair of vertical angles. Does it appear that vertical angles are congruent? Explain. Chase thinks that h and u are vertical angles. Is he correct? Explain. 15. Name a pair of adjacent complementary angles. Do you think complementary angles must be adjacent? Explain. Rosa thinks that that e and k are complementary angles. Is she correct? Explain. 16. Name a pair of supplementary angles. Do you think supplementary angles must be adjacent? Explain. 7-CORE3 SP10
3.2 Angle Explorations A f g 90 h D 65 THE TRESTLE BRIDGE A trapezoid is a parallelogram with at least one pair of parallel sides. An isosceles trapezoid is a trapezoid with a pair of congruent sides and two pairs of congruent angles (called base angles). Triangles are often used to make structures stronger. Below is a diagram of a trestle bridge that can support trains. This bridge is an isosceles trapezoid. ABCD is a rectangle. a c b d Find the measure of each angle in any order and write them in the diagram above. Then support each answer below with a short explanation or a calculation. The diagram may not be to scale. 1. m a = 2. m b = 3. m c = 4. m d = 5. m e = 6. m f = 7. m g = 8. m h = e 25 B C 50 7-CORE3 SP11
3.2 Angle Explorations USING ALGEBRA TO FIND ANGLE MEASURES 1. Write the measure of each angle. Explain your reasoning. a. m f = b. m g = 2. Write two different equations representing angle relationships shown in the diagram that include the expression 2n + 13. Then solve for n in each equation. Equation: Equation: n = 2n + 13 = n = 2n + 13 = Check: For problems 1-2 For problem 3 (2n + 13) f g 83 Diagrams may not be to scale. Check: 3. Write an equation representing the angle relationship shown in the diagram on the upper right, and solve for p. p = 2p + 4 = 3p 6 = (2p + 4) Check: (3p 6) 72 7-CORE3 SP12
3.2 Angle Explorations PRACTICE FINDING ANGLE MEASURES Write equations and solve. The diagram may not be to scale. 1. Find x. x = 2. Find y. y = Find the measure of each angle and support each answer with an explanation or calculation. 3. m a = 4. m b = 5. m c = 6. m d = 7. m e = 8. m f = (2y 10) b 50 (x 60) 90 d e c f a b 7-CORE3 SP13
3.3 Drawing Geometric Figures DRAWING GEOMETRIC FIGURES Ready (Summary) We will classify triangles based on the measure of their sides and angles. We will draw figures freehand, using rulers and protractors, and with technology. We will observe conditions that make a triangle unique. C Go (Warmup) Use the diagram for the problems below. ABGF is a rectangle. 1. Name two pairs of adjacent angles. and and 3. Name two pairs of complementary angles. and and A F D Set (Goals) Draw figures freehand, with a ruler and protractor, and using technology. Construct triangles given side lengths and angle measures. Recognize when conditions determine a unique triangle, more than one triangle, or no triangle. 2. Name two pairs of vertical angles. and and 4. Name two pairs of supplementary angles. and B G and E 7-CORE3 SP14
3.3 Drawing Geometric Figures TRIANGLE AND QUADRILATERAL REVIEW Sides Names An equilateral triangle is a triangle with three congruent sides. An isosceles triangle is a triangle with at least two congruent sides. A scalene triangle is a triangle with no congruent sides. Triangles Angle Names An acute triangle is a triangle with three acute angles. A right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. Use one word from each triangle names list to classify each triangle. If sides or angles appear to be equal in measure, assume they are. 1. 2. Quadrilaterals A trapezoid is a quadrilateral with at least one pair of parallel sides. 3. A parallelogram is a quadrilateral with two pairs of congruent, parallel sides. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides. A square is a rectangle with four congruent sides. Use at least two quadrilateral names to classify each figure. If sides or angles appear to be equal in measure, assume they are. 4. 5. 6. 7-CORE3 SP15
3.3 Drawing Geometric Figures SKETCHING FIGURES Draw each figure freehand. Do not use rulers or protractors. 1. A square with side lengths that measure 7 units each. 3. An isosceles trapezoid with a height of 4 units and bases that measure 3 units and 5 units. 2. An isosceles right triangle with the two congruent sides measuring 6 units each. 4. An obtuse scalene triangle with a height and a base that measure 4 units each. 5. Compare your figures with those drawn by classmates. Did you draw the same figures? Explain. 7-CORE3 SP16
3.3 Drawing Geometric Figures PROTRACTOR AND RULER DRAWINGS Your teacher will read directions for drawing triangles. Listen carefully and use the space below to draw your triangles with a centimeter ruler and a protractor. 1. Triangle names: Do these directions provide for a unique triangle? 2. Triangle names: Do these directions provide for a unique triangle? 7-CORE3 SP17
3.3 Drawing Geometric Figures 3. PROTRACTOR AND RULER DRAWINGS (Continued) Triangle names: Do these directions provide for a unique triangle? 4. For each triangle you drew above, measure and label all of the sides and angles. 5. Under what conditions do you think you can draw a unique triangle? Explain. 7-CORE3 SP18
3.3 Drawing Geometric Figures PRACTICE DRAWING GEOMETRIC FIGURES 1. Each of the angles in an equilateral triangle measures. Draw an equilateral triangle XYZ with sides measuring 4.5 cm each. 2. An isosceles triangle that is not equilateral has congruent sides and congruent angles. Draw an isosceles triangle with UVW measuring 40 and congruent sides UV = VW = 3.5 cm. 3. An right trapezoid that is not a parallelogram has sides, of which are parallel. Draw right trapezoid QRST with bases QR and TS measuring 5 cm and 7 cm, respectively, and a height of 2 cm. Measure the angles and write them inside the figure. 7-CORE3 SP19
3.3 Drawing Geometric Figures IMPOSSIBLE TRIANGLES The following facts may help answer questions 7-10 below. 1. The sum of the interior angles of any triangle is. 2. A(n) angle measures exactly 90 o. 3. A(n) angle measures more than 90 o, but less than 180 o. 4. An equilateral triangle has congruent sides and congruent angles. 5. A scalene triangle has congruent sides and congruent angles. 6. An isosceles triangle has congruent sides and congruent angles. Explain why it is impossible to create each triangle described below. Use diagrams to support your answers. 7. A triangle with two right angles: 8. An equilateral triangle with an obtuse angle: 9. An isosceles triangle with three different angle measures: 10. A scalene triangle with one obtuse angle and one right angle: 7-CORE3 SP20
3.4 Modeling with Expressions MODELING WITH EXPRESSIONS Ready (Summary) We will write numerical expressions to represent geometric patterns, describe patterns in words, and generalize using variable expressions. Go (Warmup) Set (Goals) Describe geometric patterns numerically, pictorially, symbolically, and verbally. Interpret expressions in terms of their geometric context. Match each variable expression in column A to its equivalent in column B. Space is provided to show work as needed. Column A 1. 3(n + 7) a. 6(n 1) 2. 2n + 6 + 3n + 5 b. 2n + 20 + n + 1 3. 4(n + 6) 5 c. 2(n + 8) + 2n + 3 4. 3(n 2) + 3n d. 5(n + 1) + 6 Column B 7-CORE3 SP21
3.4 Modeling with Expressions THE BORDER CHALLENGE 1. Other than simply counting all the way around, list at least two different ways you can think of to count the number of shaded border squares for a 12 x 12 grid. Write a numerical expression for each. 7-CORE3 SP22
3.4 Modeling with Expressions THE BORDER CHALLENGE (Continued) 2. Based on the expressions generated for problem 1, list at least three different ways to count the number of shaded border squares for an 8 x 8 grid. Write a numerical expression for each. 3. Based on the expressions generated for problems 1-2, list at least three different ways to count the number of shaded border squares for a 4 x 4 grid. Write a numerical expression for each. 4. Based on the expressions generated for problems 1-3, list at least three different ways to count the number of shaded border squares for an n x n grid. Write a variable expression for each. 7-CORE3 SP23
3.4 Modeling with Expressions ANALYZING THE BORDER CHALLENGE 1. Turn your paper to landscape orientation to complete this table. List all of the different expressions that the class generated for each square grid. Organize the expressions so that the related ones from each grid size are in the same row. Then write each expression in simplest form. n x n grid 4 x 4 grid 8 x 8 grid 12 x 12 grid a b 2. What do you notice about all of the expressions for each grid? c d e f g 7-CORE3 SP24
3.4 Modeling with Expressions ANOTHER BORDER PROBLEM 1. Two different grids are posed here. Below each grid, list all of the different ways you can think of to count the number of shaded border squares. If possible, write a numerical expression for each. 9 x 10 grid 7 x 8 grid 2. List all of the different ways you can think of to count the number of shaded border squares for a 4 x 5 grid. If possible, write a numerical expression for each. 3. Consider the established pattern of grids above. If the length of the shorter side has a length of n, then the longer side has a length of ( + ). List all of the different ways you can think of to count the number of shaded border squares for this grid. If possible, write a variable expression for each. 7-CORE3 SP25
3.5 Vocabulary, Skill Builders, and Review VOCABULARY, SKILL BUILDERS, AND REVIEW FOCUS ON VOCABULARY Word Bank for problems 1-9 straight acute right adjacent supplementary Choose the best word to describe each angle. 1. 2. 3. 4. Write ALL words that apply to each pair of angles below. 5. angles a and b: 6. angles a and c: 7. angles d and e: 8. angles e and f: 9. angles e and c: obtuse complementary vertical Diagram for problems 5-9 (may not be to scale) f d g e b c a 7-CORE3 SP26
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 First state whether each angle appears to be acute, right, obtuse, or straight. Then use a protractor to find and record the measure. 1. 2. 3. 4. First state whether each measure represents an acute, right, straight, or obtuse angle. Then use a protractor to draw the angle. One ray of the angle is given. 5. 30 7. 130 6. 65 8. 100 7-CORE3 SP27
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 Without using a protractor, write in the measure of the missing angle in each figure. Drawings may not be to scale. 1. 2. 3. 4. 5. 6.? 68 68? 7. 8. 20? 84 49 25?? 61 120? 65 75 105 73?? 75 7-CORE3 SP28
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Drawings may not be to scale. Find the measure of the missing angle in each figure. Do not use a protractor. State whether the angle is acute, right, or obtuse. 1. m a = ; 2. m b = ; 3. m c = ; 4. m d = ; 70 c a 70 5. m e = ; 6. m f = ; e 102 78 78 38 d 72 f 45 108 72 b 48 7-CORE3 SP29
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Fill in the blanks for problems 1-5, and then use these geometry facts as needed to find angle measures in the figures below. Diagrams may not be to scale. 1. The sum of the measures of the interior angles in a triangle is. 2. A straight angle measures. 3. angles are two angles with measures that total 90. They may or may not be adjacent. 4. angles are two angles with measures that total 180. They may or may not be adjacent. 5. angles are congruent, non-adjacent, opposing angles that are formed when two lines intersect. 6. m a = 7. m b = 53 8. m c = m d = 9. m e = m f = 10. m g = c m h = m k = d a k h b 120 g 120 o b e f 30 7-CORE3 SP30
3.5 Vocabulary, Skill Builders, and Review a Figure 1 Figure 2 b c d SKILL BUILDER 5 Write equations and solve for the unknowns. Diagrams may not be to scale. 1. Use figure 1 to find n in the expression below. angles a and b are complementary m a = 60 m b = (2n + 20) 3. Use figure 3 to find x and the measures of the unknown angles. x = 3x = x + 10 = (5y + 34) Figure 3 (x + 10) 2. Use figure 2 to find p in the expression below. angles c and d are supplementary m d = 65 m c = (3p 12) 4. Use figure 3 to find y and the measure of the unknown angle. y = 5y + 34 = (3x) 24 7-CORE3 SP31
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Sketch each figure freehand. Write the most specific name(s) for each figure. 1. A triangle with two congruent sides. 2. A triangle with an obtuse angle and no congruent sides. 3. A triangle with one right angle and two congruent sides. 5. A quadrilateral with four congruent sides and four congruent angles. 7. A quadrilateral with four congruent sides but not four congruent angles. 4. A triangle with three different acute angles. 6. A quadrilateral with four congruent angles but not four congruent sides. 8. A quadrilateral with exactly one pair of parallel sides. 7-CORE3 SP32
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 Use a centimeter ruler and a protractor to draw each triangle. 1. An equilateral triangle ABC with sides measuring 5 centimeters each. 3. An isosceles triangle with HJK measuring 30 and two congruent sides, HJ = JK = 4 cm. 2. A rectangle DEFG with a base of 3 centimeters and a height of 7 centimeters. 4. A rhombus PQRS with sides measuring 3 centimeters each and m SPQ = m SRQ = 50. 7-CORE3 SP33
3.5 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 1. Two different grids are posed here. Below each grid, list all of the different ways you can think of to count the number of shaded border squares. If possible, write a numerical expression for each. 8 x 10 grid 6 x 8 grid 2. List all of the different ways you can think of to count the number of shaded border squares for a 3 x 5 grid. If possible, write a numerical expression for each. 3. Consider the established pattern of grids above. If the length of the shorter side has a length of n, then the longer side has a length of ( + ). List all of the different ways you can think of to count the number of shaded border squares for this grid. If possible, write a variable expression for each. 7-CORE3 SP34
3.5 Vocabulary, Skill Builders, and Review TEST PREPARATION Show your work on a separate sheet of paper and choose the best answer. Diagrams may not be to scale. 1. Choose all words that appear to describe the triangle. A. acute B. obtuse C. isosceles D. scalene 2. What is the measure of the missing angle in the figure? A. 103 B. 114 C. 180 D. 360 3. Choose all equations that can be used to solve for x. A. x + 10 + 40 = 90 B. x + 10 + 40 = 180 C. x + 10 + 40 + 90 = 180 D. x + 50 = 180 4. Choose all words that describe a and b as a pair of angles. A. adjacent B. complementary C. supplementary D. vertical 5. Choose all descriptions that CANNOT describe an actual figure. A. A triangle with more than one right angle. C. A triangle with more than one acute angle. 2cm 87 82 2cm 77? (x+10) B. A triangle with more than one obtuse angle. D. A rhombus with one pair of opposite sides longer than the other pair of opposite sides. a 40 b 7-CORE3 SP35
3.5 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK Show work on a separate sheet of paper as needed. Write answers on this page. Diagrams may not be to scale. 3.1 Geometric Figures Draw angles with the following measures. 1. 90 2. 180 3. 40 4. 135 Find the measures of the missing angle in each figure below without a protractor. 5. 6. 7. 136 3.2 Angle Explorations? Write two pairs of each of the following types of angles that appear in this figure. 8. Adjacent angles 9. Vertical angles 10. Complementary angles 11. Supplementary angles 3.3 Drawing Geometric Figures Use a ruler and protractor to draw the following. 12. An equilateral triangle with sides that measure 4.5 cm. 3.4 Modeling With Expressions 14. A rectangle has dimensions n and (n + 3). Write more than one expression to represent the number of border tiles for this rectangle. 51 86 13. A rhombus ABCD with sides lengths of 4 cm, m A = m B = 45 o and m C = m D = 135 o....... 121?? 59 59 D E F G A B... C 7-CORE3 SP36
3.5 Vocabulary, Skill Builders, and Review HOME-SCHOOL CONNECTION Here are some questions to review with your young mathematician. 1. Sketch an isosceles right triangle ABC with congruent sides AB and BC. 2. Use this figure to find the missing angles measures. Explain your reasoning. The diagram may not be to scale. m a = m c = m e = m g = m k = n h k m b = m d = m f = m h = m n = Parent or Guardian Signature g 48 b c a 24 f d e 7-CORE3 SP37
COMMON CORE STATE STANDARDS MATHEMATICS 4.G.1 4.G.2 4.MD.5a 4.MD.5b 4.MD.6 4.MD.7 6.EE 3 6.EE 4 6.EE 6 7.EE 1 7.EE 2 STANDARDS FOR MATHEMATICAL CONTENT Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. 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. 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. 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. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that increase by 5% is the same as multiply by 1.05. 7.G 2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 7.G 5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. MP1 MP2 MP3 MP4 MP5 MP6 MP7 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. 2013 Center for Mathematics and Teaching, Inc. 7-CORE3 SP38