NPW Grade 7- Teacher Edition Sample_clone

Arc of Learning, Standards, Now What Do You Know?, and Emerging Mathematical Ideas The detailed description of the Arc of Learning (AoL) can be found in A Guide to Connected Mathematics® 4 and the online portal. Unit standards correlations can be found at the end of this Teacher Edition or on the Lab-Aids website. Problem 1.1. Designing Triangles Experiment: The Side Connection Arc of Learning: Introduction, Exploration, Introduction Standards: See your state alignment chart. Now What Do You Know? What combinations of three side lengths can be used to make a triangle? How many different triangles are possible for each combination? Emerging Mathematical Ideas Analyze triangles looking for relationships among side lengths that will or will not form triangles. Notice that • for every triangle, the sum of two side lengths will always be greater than the length of the third side; • given a set of three side lengths, if one of the sides is not longer than the sum of the other two sides, a triangle cannot be formed; and • for every set of three side lengths that can be used to form a triangle, there is only one triangle that can be made. Use problem contexts with triangles to explore, hypothesize, draw conclusions, and solve problems. Problem 1.2. Designing Quadrilaterals Experiment: The Side Connection Arc of Learning: Exploration, Introduction, Exploration Standards: See your state alignment chart. Now What Do You Know? What do we know about sidelength relationships among the sides of a polygon? Emerging Mathematical Ideas Analyze quadrilaterals looking for relationships among side lengths and for properties that might or might not make them useful in construction applications. Look for and compare side-length relationships for triangles, quadrilaterals, pentagons, and hexagons. Notice that when forming quadrilaterals • the measure of every side must be smaller than the sum of the measures of the other three sides; and • many quadrilaterals can be made from any set of four sides lengths that meet the above requirement. Notice that when force is applied to one vertex of a shape • triangles are rigid—once formed, the angles are fixed, creating a nonflexible framework; • quadrilaterals shift easily under pressure and are not stable in the way triangles are; and • like quadrilaterals, pentagons and hexagons are also not stable under pressure unless you add diagonals that create rigid triangles inside the shape. Comparing side-length relationships for triangles, quadrilaterals, and other polygons, notice that and explain or demonstrate why • for both triangles and quadrilaterals, the length of any side will be less than the sum of the lengths of the other sides; and Unit Planning UP-19 SAMPLE

• like triangles and quadrilaterals, for pentagons and hexagons the length of any one side must be less than the sum of the lengths of the other sides in order to form a closed polygon. Use problem contexts with quadrilaterals to explore, hypothesize, draw conclusions and solve problems. Problem 1.3. Rigidity Experiment Arc of Learning: Introduction, Exploration, Introduction, Exploration Standards: See your state alignment chart. Now What Do You Know? What do you know about the rigidity of shapes, and how does this explain the frequent use of triangles in building structures? Emerging Mathematical Ideas Analyze polygons and diagonals looking for rigidity. Find the relationship among the number of sides and the minimum number of diagonals to make a polygon rigid. Notice that when adding diagonals to polygons • the minimum number of diagonals to make a polygon rigid is the number of sides minus 3; and • adding the minimum number of diagonals to a shape creates embedded triangles. Generalize the relationship between the number of sides of a polygon and the number of angles needed to make the shape rigid. Mathematical Reflection Arc of Learning: Exploration, Exploration Standards: See your state alignment chart. Mathematical Reflection What do you know about geometric shapes? Emerging Mathematical Ideas Understand and use generalized relationships among side lengths of polygons to form triangles and quadrilaterals and to solve problems including • for both triangles and quadrilaterals, the length of any side will be less than the sum of the lengths of the other sides; • for every set of three side lengths that will form a triangle, there is only one triangle that can be formed; • for every set of four side lengths that will form a quadrilateral, several different quadrilaterals can be made from the set of sides lengths; • triangles make strong frameworks for buildings, as they are rigid under pressure, and since quadrilaterals can shift easily under pressure, they are less useful as frameworks for buildings; • the minimum number of diagonals to make a polygon rigid is the number of sides minus 3; and • adding the minimum number of diagonals to a shape creates embedded triangles. UP-20 Unit Planning SAMPLE

Investigation 2: Designing with Angles Goals Generalizing and Using Properties of Polygons. Understand the properties of polygons that affect their shape and how this information is useful in solving problems. • Investigate techniques for estimating, measuring, and sketching angles and recognizing the effects of measurement accuracy • Recognize and use information about supplementary, complementary, vertical, and adjacent angles to solve problems • Reason about the properties of angles formed by intersecting lines and by parallel lines cut by a third line and how this information relates to polygons • Explore the relationship between interior and exterior angles of a polygon • Explore the relationships between angle measures, angle sums, and the number of sides in a polygon • Determine which polygons fit together to cover a flat surface and why • Draw or sketch polygons with given conditions using various tools and techniques, such as freehand, geoboards, use of a ruler and protractor, and use of technology • Determine what conditions will produce a unique polygon, more than one polygon, or no polygon, particularly triangles and quadrilaterals • Recognize the special properties of polygons that make them useful in building, design, and nature • Solve multistep problems that involve properties of shapes Algebraic Expressions and Equations. Understand how expressions and equations can be useful to express geometric relationships and how this information is used to solve problems. • Use algebraic equations or expressions to represent geometric patterns and solve problems • Recognize that equivalent expressions can reveal different information about a situation and how the quantities are related Unit Planning UP-21 SAMPLE

Arc of Learning, Standards, Now What Do You Know?, and Emerging Mathematical Ideas The detailed description of the Arc of Learning (AoL) can be found in A Guide to Connected Mathematics® 4 and the online portal. Unit standards correlations can be found at the end of this Teacher Edition or on the Lab-Aids website. Problem 2.1. Four in a Row Game: Angles and Rotations Arc of Learning: Exploration, Analysis, Exploration Standards: See your state alignment chart. Now What Do You Know? How are benchmark angles useful in solving problems? Emerging Mathematical Ideas Begin making sense of reading, sketching, and problem-solving with benchmark angles described by the number of degrees or the fraction of a turn (rotation). Recognize and sketch these benchmark angles: • 30° or __1 12 of a turn • 45° or __1 8 of a turn • 90° or __1 4 of a turn • 120° or __1 3 of a turn • 180° or __1 2 of a turn • 270° or __3 4 of a turn • 360° or 1 full turn Solve and make sense of equations involving angles such as __1 4 N = 360°, where N is the angle measure of a rotation angle. Problem 2.2. The Bee Dance and Amelia Earhart: Measuring Angles and Distance Arc of Learning: Exploration, Analysis Standards: See your state alignment chart. Now What Do You Know? What are the advantages and disadvantages of estimating angle measures? What are the advantages and disadvantages of using a protractor or angle ruler to measure an angle? Emerging Mathematical Ideas Extend understanding and applications of angle measure to real-world problems, and explore measurement limitations and possible effects of estimation and/or measurement error. Explore the honeybee and navigation contexts, noticing that • since no measuring tool is perfectly precise, no measure is exact and every measure has some error; • small errors in the measure of an angle can result in very large errors the farther you get from the angle center going out along an angle ray; and • estimating the measure of the angle offers students a way to check the values they get using a measurement tool. Use the “navigation during flight” context to • identify variables in the real-world situation; • look for a proportional relationship in the table of values and use it to describe the relationship as an equation; and • use the equation to answer questions about the navigation context. UP-22 Unit Planning SAMPLE

Problem 2.3. Vertical, Supplementary, and Complementary Angles Arc of Learning: Exploration, Analysis, Analysis Standards: See your state alignment chart. Now What Do You Know? Describe what you know about angles formed by intersecting lines. Include parallel lines and the angles formed by a line that intersects the parallel lines in your description. Emerging Mathematical Ideas Extend understanding and applications of angle measure to include angles created by two intersecting lines or by two parallel lines cut by a third intersecting line. Explore, identify, and describe angle relationships created by intersecting lines, including • adjacent angles, which are two angles that share a common endpoint and a common ray (side); • vertical angles that are formed when two lines intersect share a common endpoint, do not share a common ray (side), and have equal measures; • supplementary angles, which are a pair of angles whose sum is 180°; • complementary angles, which are a pair of angles whose sum is 90°; and • parallel lines are lines that never intersect and are always the same distance apart. Use the angle relationships above to write equations and solve multistep problems involving angles formed with intersecting lines. Mathematical Reflection Arc of Learning: Exploration, Analysis Standards: See your state alignment chart. Mathematical Reflection What do you know about geometric shapes? Emerging Mathematical Ideas Recall, deepen, and extend elementary school experiences with angle and angle measure to include a variety of angle relationships, benchmark angles, contexts with angles other than geometric shapes, and problem-solving with angle measures. Understand and make use of key angle concepts and skills for problemsolving, including • read, sketch, estimate, and use benchmark angles described by the number of degrees or the amount of turn; • understand measurement limitations and possible effects of estimation and/or measurement error; • apply angle concepts and skills to both mathematical problem contexts such as intersecting lines (including parallel lines) and real-world contexts beyond geometric shapes; • recognize, describe, and use angle relationships such as complementary and supplementary angle pairs, adjacent angles, vertical angles, and straight angles; and • begin to use angle relationships to write equations and solve multistep problems involving angles. Unit Planning UP-23 SAMPLE

Investigation 3: Designing Polygons: The Angle Connection Goals Generalizing and Using Properties of Polygons. Understand the properties of polygons that affect their shape and how this information is useful in solving problems. • Investigate techniques for estimating, measuring, and sketching angles and recognizing the effects of measurement accuracy • Recognize and use information about supplementary, complementary, vertical, and adjacent angles to solve problems • Reason about the properties of angles formed by intersecting lines and by parallel lines cut by a third line and how this information relates to polygons • Explore the relationship between interior and exterior angles of a polygon • Explore the relationships between angle measures, angle sums, and the number of sides in a polygon • Determine which polygons fit together to cover a flat surface and why • Draw or sketch polygons with given conditions using various tools and techniques, such as freehand, geoboards, use of a ruler and protractor, and use of technology • Determine what conditions will produce a unique polygon, more than one polygon, or no polygon, particularly triangles and quadrilaterals • Recognize the special properties of polygons that make them useful in building, design, and nature • Solve multistep problems that involve properties of shapes Algebraic Expressions and Equations. Understand how expressions and equations can be useful to express geometric relationships and how this information is used to solve problems. • Use algebraic equations or expressions to represent geometric patterns and solve problems • Recognize that equivalent expressions can reveal different information about a situation and how the quantities are related UP-24 Unit Planning SAMPLE

Arc of Learning, Standards, Now What Do You Know?, and Emerging Mathematical Ideas The detailed description of the Arc of Learning (AoL) can be found in A Guide to Connected Mathematics® 4 and the online portal. Unit standards correlations can be found at the end of this Teacher Edition or on the Lab-Aids website. Unit Planning UP-25 Problem 3.1. Back to the Bees: Tiling a Plane Experiment Arc of Learning: Analysis, Synthesis Standards: See your state alignment chart. Now What Do You Know? Which regular polygons can be used to tile a surface? Explain why they tile. Give some examples of how tiling polygons can be useful. Emerging Mathematical Ideas Analyze and solve geometric problems focused on polygons in the context of tiling to deepen understanding of polygon angle characteristics and relationships. Explore and solve geometric problems, applying knowledge of polygon angles, including • show that some but not all polygons will tile a surface (triangle, quadrilateral, and hexagon), noting that they will fit snuggly into each other without gaps or overlaps; • analyze why the shapes will or will not tile, thinking about the measure of the shape’s angles to notice that shapes that tile have angle measures that add to 360°; • create shape patterns that tile with two or more regular polygons and be prepared to explain why they will tile based on angle sums in the pattern that equal 360°; and • give examples of real-life ways in which shapes are used to tile/cover surfaces, such as the bee’s honeycomb or an old-fashioned quilt. Problem 3.2. Relating Angle Measures to Number of Sides of Polygons Experiment Arc of Learning: Analysis, Synthesis, Analysis Standards: See your state alignment chart. Now What Do You Know? What is the relationship between the angle sum S of a polygon with n sides and the number of sides? How can you find the measure of an angle in a regular polygon with n sides? Emerging Mathematical Ideas Explore side (n) and angle (A) relationships for polygons, and develop and apply equations relating the number sides with the sum of the angles (S). Explore, draw conclusions, and apply angle and side relationships in polygons to solve problems, including • measure angles to complete a table for regular polygons listing the number of sides (n), measure of one inside angle (A), and the sum of the angles; • analyze patterns in the table to look for side/angle relationships that can be described in words (e.g., the sum of the angle measures inside a polygon is equal to the number of sides minus 2, then multiplied by 180°, or you can find the value of one angle in a regular polygon by dividing the sum of the inside angles by the number of angles [same as number of sides]); • create and make connections among equivalent equations for calculating the sum of the inside angles of a polygon based on patterns found in the table and geometric strategies that partition shapes into triangles, for example, S = 180°(n − 2) S = 180n − 360°; SAMPLE

• note that the side/angle relationship is the same for regular, irregular, concave, and convex polygons; • for regular polygons, make sense of using the sum of the angle measures to find the measure of one angle by dividing the total by the number of angles (sides) in the polygon, S = [180° * (n − 2)] ____________ n ; and • use these equations, knowledge of angle relationships, and partitioning strategies to solve multistep problems. Problem 3.3. The Ins and Outs of Polygons: Using Supplementary Angles Arc of Learning: Analysis, Synthesis, Analysis Standards: See your state alignment chart. Now What Do You Know? What do you know about the exterior angles of a polygon? How might this knowledge be useful? Emerging Mathematical Ideas Analyze and draw conclusions about the sum of the exterior angles of any polygon and the relationship between the sums of exterior angles and the sum of interior angles. Analyze relationships in a variety of polygons to notice • while the sum of the interior angles of polygons changes depending on the numbers of sides, the sum of the exterior angles of any polygon is always 360°; • each pair of interior and exterior angles on a polygon forms a straight angle (or are supplementary or sum to 180°); • using the observation above, you can find the total sum of interior and exterior angles of a polygon by multiplying the number of sides (n) times 180°; • the total sum of interior and exterior angles can be used as another way to find the sum of the interior angles (S), that is, S = 180°n − 360°; and • use these equations, knowledge of tiling, and angle relationships (supplementary, complementary, vertical) to solve multistep problems. Problem 3.4. Designing Polygons Arc of Learning: Synthesis, Abstraction, Analysis Standards: See your state alignment chart. Now What Do You Know? What information about a triangle allows you to draw a triangle? Draw a unique triangle? Explain why this is true. What information about a quadrilateral allows you to draw a quadrilateral? Explain why this true. Emerging Mathematical Ideas Analyze sets of triangle and quadrilateral criteria looking for minimum combinations of sides and/or angles that will describe a unique polygon. Begin to notice and generalize the following minimal combinations of sides and/or angles that will describe a single unique triangle: • the measures of two sides and the included angle • the measures of two angles and one side • the measure of three sides Other combinations of criteria will allow for either multiple triangles (AAA) to be formed or no triangle to be formed (e.g., 3 inches, 5 inches, and 10 inches). Analyze sets of criteria (angle and side measures expressed with numbers or variables) and draw conclusions using the following facts about triangles and quadrilaterals: • The sum of the angle measures of a triangle must equal 180°. • The sum of the angle measures of a quadrilateral must equal 360°. • The measure of every side of a triangle or a quadrilateral must be less than the sum of the other three sides. UP-26 Unit Planning SAMPLE

Unit Planning UP-27 Mathematical Reflection Arc of Learning: Analysis, Synthesis, Analysis Standards: See your state alignment chart. Mathematical Reflection What do you know about geometric shapes? Emerging Mathematical Ideas Further analyze angle relationships in regular polygons, and in concave and convex polygons to include tiling, and patterns for finding the sum of interior and exterior angles. Notice that when forming triangles and quadrilaterals • the measure of every side must be less than the sum of the measures of the other three sides; • only one triangle can be made from three side lengths that meet the criteria above; and • many quadrilaterals can be made from any set of four sides lengths that meet the above requirement. Notice that when force is applied to one vertex of a shape • triangles are rigid—once formed, the angles are fixed, creating a nonflexible framework; • quadrilaterals shift easily under pressure and are not stable in the way triangles are (note: this flexibility is useful in applications where shapes need to collapse); and • like quadrilaterals, pentagons and hexagons are also not stable under pressure unless you add diagonals that create rigid triangles inside the shape. Summarize and use angle relationships in polygons and intersecting lines, including • the sum (S) of the interior angles of a polygon can always be found using the formula S = 180 × (n − 2) (n is the number of sides) or S = 180n − 360; • the sum of the exterior angles of any polygon is always 360°; • each pair of interior and exterior angles on a polygon forms a straight angle (or are supplementary or sum 180°); • for regular polygons, the measure of one interior angle can be found by finding the sum (S) of the interior angles of a polygon with n sides using the formula S = 180 × (n − 2) and then dividing by the number of angles (sides) to find the measure of one angle, • some but not all of the regular polygons will tile a surface (triangle, square, and hexagon); • understand that the reason why some regular polygons or combinations of polygons will tile is that these shapes have angles that are factors of 360°; • benchmark angles can help to estimate angle size or check accuracy when using a measuring tool; • two supplementary angles sum to 180°; • two complementary angles sum to 90°; and • vertical angles have equal measures. Use problem contexts with triangles and quadrilaterals to explore, hypothesize, draw conclusions, and solve problems. Begin to recognize and use vocabulary, including • regular polygons; • interior and exterior angles in polygons; • supplementary angles and complementary angles; and • vertical angles. SAMPLE

© 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. UP-28 Unit Planning Dear Family, The first unit in your student’s mathematics class this year is Shapes and Designs: Generalizing and Using Properties of Geometric Shapes. Students will recognize, analyze, measure, and reason about the shapes and visual patterns that are important features of our world. It builds on students’ elementary school exposure to simple shapes by analyzing the properties that make certain shapes special and useful. Students will also revisit and build on their work in grade 6 with algebraic expressions and equations to represent geometric patterns and solve problems. Unit Goals The goal is to have students discover and analyze key properties of polygonal shapes that make them useful and attractive. This unit has two major mathematical goals: Generalizing and Using Properties of Polygons. Understand the properties of polygons that affect their shape and how this information is useful in solving problems. Algebraic Expressions and Equations. Understand how expressions and equations can be useful to express relationships and how this information is used to solving problems. This unit focuses on polygons and develops two basic subthemes: 1. What determines the unique shape of a polygon? 2. What properties of angles and polygons make them useful in design, building, and nature? Homework and Having Conversations About the Mathematics You can help with homework and encourage sound mathematical habits during this unit by asking questions such as: • How do simple polygons work together to make more complex shapes? • How can angle measures be estimated? How can angles be measured with more accuracy? • What kinds of shapes/polygons will cover a flat surface? You can help your student with their work for this unit in several ways: • Whenever you notice an interesting shape, discuss with your student whether it is one of the polygons mentioned in the unit. Ask your student about shapes that they find interesting. • Have your student share their mathematics notebook with you. Ask about what has been recorded about the shapes being studied. Ask about the relationships or patterns that they have learned. Ask your student to explain why these ideas are important, and try to share ways that shapes help you with work or hobbies. • Look over your student’s homework, and make sure all questions are answered and that explanations are clear. A few important mathematical ideas that your student will learn are on the following page. As always, if you have any questions or concerns, please feel free to contact me. All of us here are interested in your student and want to be sure that this year’s mathematics experiences are enjoyable and promote a firm understanding of mathematics. Sincerely, SAMPLE

Important Concepts Examples Polygon Polygons are two-dimensional shapes formed by linking points called vertices with line segments called sides. vertex side Polygons Polygon Names Triangle: 3 sides and 3 angles Quadrilateral: 4 sides and 4 angles Pentagon: 5 sides and 5 angles Hexagon: 6 sides and 6 angles Heptagon: 7 sides and 7 angles Octagon: 8 sides and 8 angles Nonagon: 9 sides and 9 angles Decagon: 10 sides and 10 angles Dodecagon: 12 sides and 12 angles Angles Angles are figures formed by two rays or line segments that have a common vertex. The vertex of an angle is the point where the two rays meet or intersect. Angles are measured in degrees. B A P B A P Angle Measures Work is done to relate angles to right angles, to develop students’ estimation skills. Combinations and partitions of 90° are used. 30°, 45°, 60°, 90°, 120°, 180°, 270°, and 360° are used as benchmarks to estimate angle size. 90° 270° 45° 60° 120° 180° The need for more precision requires techniques for measuring angles. Students use a goniometer (goh nee AHM uh tur), or angle ruler. This is a tool used in the medical field for measuring angle of motion or the flexibility in body joints, such as knees. 0 1 2 3 4 5 6 0 1 2 3 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 100 80 110 120 130 140 150 160 170 170 160 150 140 130 120 110 100 80 90 70 60 50 40 30 20 10 Angle Ruler Center Line Center Line Rivet The angles on opposite sides of a vertex are called vertical angles. 4 1 3 2 4 1 3 2 ∠4 and ∠2 ∠1 and ∠3 Two angles that have a common side and a common vertex and don’t overlap are adjacent angles. 4 1 3 2 4 1 3 2 ∠1 and ∠2 ∠2 and ∠3 Shapes and Designs © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Unit Planning UP-29 SAMPLE

Important Concepts (continued) Examples (continued) Two angles are supplementary if the sum of their measures is 180º. If they are adjacent, they form a straight line. 60° 60° 30° 30° 150° 30° 60° 60° 30° 30° 150° 30° Two angles are complementary if the sum of their measures is 90º. If they are adjacent, they form a right angle. 60° 60° 30° 30° 150° 30° 60° 60° 30° 150° 30° 30° Triangle Inequality Theorem Students find that the sum of two side lengths of a triangle must be greater than the third side length. If the side lengths are a, b, and c, then the sum of any two sides is greater than the third: a + b > c, b + c > a, c + a > b (continued from page UP-29) © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. UP-30 Unit Planning SAMPLE

Implementation Key Terms Materials Resources Problem 1.1 Groups of 2–3 Pacing 1 day polygons regular polygons For each student • Learning Aid 1.1: Building Triangles For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips (optional) Problem 1.2 Groups of 2–3 Pacing 1 day For each student • Learning Aid 1.2: Building Quadrilaterals For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips (optional) Problem 1.3 Groups of 2 Pacing 1 day For each student • Learning Aid 1.3A: Diagonals and Rigidity • Learning Aid 1.3B: Diagonals and Rigidity Shapes (optional) Mathematical Reflection Whole Class Pacing __1 2 day For the class • large poster paper (optional) word bank created by students and/or teacher (optional) INVESTIGATION 1 PLANNING CHART Designing Polygons: The Side Connection INVESTIGATION 1 Materials for All Investigations: calculators; student notebooks; colored pens, pencils, or markers 1 SAMPLE

At a Glance This problem develops student understanding of the basic facts that in a triangle the sum of any two sides must be greater than the third and that once three acceptable side lengths have been chosen, there is only one triangular shape with those side lengths. In the Initial Challenge, students use polystrips to find side lengths that will make triangles. The What If . . . ? situations will have them examine some specific triangles. PROBLEM 1.1 Designing Triangles Experiment: The Side Connection NOW WHAT DO YOU KNOW? What combinations of three side lengths can be used to make a triangle? How many different triangles are possible for each combination? Key Terms Materials polygons regular polygons For each student • Learning Aid 1.1: Building Triangles For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips (optional) Pacing 1 day Groups 2–3 students A 1–6 C 15–16 E 21 Arc of Learning Introduction Exploration Introduction Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE As you are going over the Looking Ahead, spend time discussing polygons and regular polygons. You might find it useful to initiate work on the investigation by asking students where they have seen triangles used in structures and then why they think those extra braces are used. Suggested Question • What is it that makes triangles so common in construction? PRESENTING THE CHALLENGE It is a good idea to demonstrate how to use the polystrips and how to make a triangle given the lengths. First, choose three numbers less than 20, such as 6, 8, 12. Build a triangle with the polystrips using those numbers as side lengths. Then have each group build a 6-8-12 triangle. Check to see that each group knows how to fasten the strips together to represent the lengths. Use Fist to Five to decide the comfort level that students have with using the polystrips before having them begin the problem. Language Fist to Five 2 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Facilitating Discourse Teacher Moves LAUNCH Once they are comfortable with how to use the polystrips, have students work in groups of 2 or 3 students to complete this problem for other triangles. To choose the lengths of sides for their possible triangles, you could have students roll a number cube. Suggested Question • Will any three side lengths make a triangle? Make a Prediction EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Make sure students produce sample sets of side lengths that can be used for constructing a triangle and other sample sets that cannot be used. Have students record the sets of side lengths that can be used to make a triangle versus those that cannot so that they may draw their own conclusions from the data collected. As students continue to explore types of triangles using polystrips, encourage them to try to find patterns in the side lengths of the triangles they find. Asking students to restate how another person describes their conjecture can help a group to decide how to write it symbolically. For What If . . . ? Situation A, some students may need some reminders of the definitions of equilateral, isosceles, and right triangles. PLANNING FOR THE SUMMARY As you are circulating, take note of the conjectures students have on the combinations of three side lengths that can be used to make a triangle. Encourage students to look for patterns in the data they have collected. Choose several examples from students that will make triangles and examples that will not to use in the summary. Restating Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES The most important result from this problem is to answer the question, “What combinations of side lengths can and cannot be used to make a triangle?” Students should be able to articulate, in reasonably clear language, the principle that the sum of any two side lengths must always be greater than the third side length. They should be able to give an argument about why that is true, accompanied by a demonstration with polystrips. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • Can we come up with a summary statement that would help someone who is not here today know how to judge whether three lengths will make a triangle without actually building the triangle? • Which sets of side lengths form a triangle? Why? • Do any of these triangles have special properties? Describe them. • For What If . . . ? Situation A, how could Geraldo make his triangular doorframe? • Do you have any What If . . . ? questions? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Claim, Support, Question Problem 1.1 Designing Triangles Experiment: The Side Connection 3 SAMPLE

Problem Overview This problem develops student understanding of the basic facts that in a triangle the sum of any two sides must be greater than the third and that once three acceptable side lengths have been chosen, there is only one triangular shape with those side lengths. The most effective use of this problem will result if you have sets of polystrips and fasteners for each group to use. Launch (Getting Started) Connecting to Prior Knowledge As you are going over the Looking Ahead, spend time discussing polygons and regular polygons. You might find it useful to initiate work on the investigation by asking students where they have seen triangles used in structures and then why they think those extra braces are used. (Language) Suggested Questions • What kinds of shapes do you see as you look at buildings and construction? (Answers will vary.) • What is it that makes triangles so common in construction? (Answers will vary.) If some students happen to know that triangles are rigid figures, you can segue into the problem by suggesting that some exploration with the construction of triangles can help explain that property. If no student suggests the fact about rigidity, you can simply suggest that building and testing the strength of some triangles will be helpful in understanding the phenomenon. Presenting the Challenge When using the polystrips to represent the length, help students to be conscientious. They need to be aware that it is the space between two holes that represents a length of one unit. We count the space (distance), not holes. For example, with fence posts, it takes three fence posts to hold up two lengths of fence. If we want to know how long the fence is, we count fence sections, not posts. You may want to number the holes starting with zero on the polystrips so students can immediately find the given length. (Portrayal) EXTENDED LAUNCH—EXPLORE—SUMMARIZE 4 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Extended Launch—Explore—Summarize 5 Distribute polystrips to groups of students. If you don’t have polystrips, you can make them using Learning Aid Template: Polystrips. (Problem-Solving Environment) It is a good idea to demonstrate how to use the polystrips and how to make a triangle given the lengths. First, choose three numbers less than 20, such as 6, 8, 12. Build a triangle with the polystrips using those numbers as side lengths. Then have each group build a 6-8-12 triangle. Check to see that each group knows how to fasten the strips together to represent the lengths. Use Fist to Five to decide the comfort level that students have with using the polystrips before having them begin the problem. Suggested Question • Will any three side lengths make a triangle? (This is a prediction, so accept all reasonable answers. They will investigate this in the experiment in the Initial Challenge.) Distribute Learning Aid 1.1: Building Triangles, and have students record the 6, 8, 12 side lengths on the learning aid. Once they are comfortable with how to use the polystrips, have students work in groups of 2 or 3 students to complete this problem for other triangles. To choose the lengths of sides for their possible triangles, you could have students roll a number cube. Explore (Digging In) Providing for Individual Needs As students work with the polystrips during the Initial Challenge, keep in mind that a key goal of the experimentation is to help them discover the Triangle Inequality Theorem (the sum of any two side lengths must always be greater than the third side length). With this in mind, make sure students produce sample sets of side lengths that can be used for constructing a triangle and other sample sets that cannot be used. Have students record the sets of side lengths that can be used to make a triangle versus those that cannot so that they may draw their own conclusions from the data collected. (Portrayal) When students address the question about whether it is possible to make more than one triangle with any three side lengths, they might say, “Yes.” What they are probably thinking is that the same triangle viewed from different perspectives is actually a different shape. You might ask them to make two triangles with corresponding sides the same length. Ask them to see if they can or cannot fit one triangle exactly on top of the other by suitable turns, slides, and flips. LES SAMPLE

LES 6 Investigation 1 Designing Triangles Experiment: The Side Connection As students continue to explore types of triangles using polystrips, encourage them to try to find patterns in the side lengths of the triangles they find. Suggested Questions • What examples have you created? (Answers will vary.) • Do any of these examples have special properties? (Answers will vary. Look for right, isosceles, and equilateral triangles.) • What combinations of sides can be used to make triangles? (Answers will vary. The sum of the two short sides is greater than the longest side.) • What is a summary statement that you could use to help someone who is not here today know how to judge whether three lengths will make a triangle without actually building the triangle? (Explanations will vary. The sum of the two short sides is greater than the longest side.) • How can you represent your conjecture symbolically? (Accept all reasonable answers) Asking students to restate how another person describes their conjecture can help a group to decide how to write it symbolically. When testing the pattern on equilateral, isosceles, and right triangles, some students may need some reminders of the definitions for the different triangles. Also, some students may have created some or all of these triangle types during the Initial Challenge. If they have done so, you can challenge them to find others or have them go on to What If . . . ? Situation A. (Agency, Identity, Ownership) Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? What combinations of three side lengths can be used to make a triangle? How many different triangles are possible for each combination? As you are circulating, take note of the conjectures students have on the combinations of three side lengths that can be used to make a triangle. Encourage students to look for patterns in the data they have collected. Choose several examples from students that will make triangles and examples that will not make triangles to use in the summary. SAMPLE

Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies The most important result from this problem is to answer the question “What combinations of side lengths can and cannot be used to make a triangle?” Students should be able to articulate, in reasonably clear language, the principle that the sum of any two side lengths must always be greater than the third side length. They should be able to give an argument about why that is true, accompanied by a demonstration with polystrips. (Language) If students have not concluded that the sum of any two side lengths must always be greater than the third side length, you can push the thinking. Collect class data in two groups: three lengths that make a triangle and three lengths that do not make a triangle. You might also model collecting the lengths in an organized way by listing the lengths in ascending order. (Portrayal) For example, a collection of some class data might look like this: Three Lengths That Make a Triangle Three Lengths That Do Not Make a Triangle 3, 3, 3 1, 2, 4 3, 4, 5 1, 5, 10 4, 6, 9 2, 2, 7 6, 10, 12 3, 5, 8 8, 9, 10 6, 7, 14 Making the Mathematics Explicit Have groups make conjectures about what lengths will and will not make a triangle and explain why. Here are some conjectures students have made: • Daniella said that two short sides added together have to be more than the longest side. • Paul said that if the two short sides are less than the long side, they fall on top of each other. • Yvonne said the two short sides could not add up to the same as the long side or they won’t stick up and leave any space inside. Suggested Questions • Can we come up with a summary statement that would help someone who is not here today know how to judge whether three lengths will make a triangle without actually building the LES Extended Launch—Explore—Summarize 7 SAMPLE

triangle? (Write the summary statement on the board. As students add to the discussion, revise the statement as improvements are suggested. Ask questions to stretch the students’ thinking until you have a rule that clearly distinguishes lengths that will work from those that won’t.) One good way to push their thinking in a situation like this is to make up examples to test. (Time) Give students the following sets of numbers: 4, 3, 5; 8, 2, 12; 8, 8, 4. • Which sets form a triangle? Why? (These numbers form triangles 4, 3, 5 and 8, 8, 4 because the sum of any two sides is greater than the third.) • Do any of these triangles have special properties? Describe them. (8, 8, 4 is isosceles since it has two equal sides. 4, 3, 5 is a right triangle because it has one 90° angle.) • For What If . . . ? Situation A, how could Geraldo make his triangular doorframe? (Have students demonstrate their answers with the Triangle Inequality Theorem or by making them with polystrips. He can make 3-3-3, 3-3-5, 5-5-6, 5-5-7, 3-5-6, 3-5-7, 3-6-7, 5-6-7 triangles for the doorframe. The most likely solution is to make two 3-3-5 frames, one for the door and one for the rear.) • When you built your triangles, were you able to use the same side lengths to make a different triangle? (No.) • Do you have any What If . . . ? questions? Continue to pose questions that ask students to explain why certain lengths work and others do not. Some teachers take this opportunity to discuss with students how mathematicians think and record the results of their experimentation. You might want to show students how mathematicians use the language of mathematics to record their generalization. Before you share, tell them that they are not responsible for knowing the following yet. Mathematicians talk about ideas at a general level rather than a specific one. For example, they name the lengths of a triangle’s sides rather than using specific side lengths such as 8 cm, 6 cm, 5 cm. They give the sides of a triangle letters for names, such as sides a, b, and c. So a triangle with sides a, b, and c stands for any triangle you can make. Then mathematicians would write a statement like this: “If a and b represent the two shorter sides of a triangle and c represents the longest side, then a + b > c.” This is called the Triangle Inequality Theorem. LES 8 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES Extended Launch—Explore—Summarize 9 SAMPLE

INITIAL CHALLENGE The best way to discover what is so special about triangles in construction is to build several models and explore the relationship among the side lengths. Use polystrips or other tools to make and study several triangles. Make a Prediction • Will any three side lengths make a triangle? Student predictions will vary based on their past experiences. Conduct the Experiment Equipment › polystrips › fasteners Directions › Step 1. Pick three numbers between 2 and 20 for side lengths. Use them as side lengths to build a triangle. › Step 2. Try to make a triangle with the chosen side lengths. If you can build a triangle, try to build a different triangle with the same side lengths. › Repeat Steps 1 and 2 to make and study several other triangles. Record your data in a table with headings similar to the one here. Side Lengths Triangle Possible? Sketch Different Triangle Possible? Analyze the Data • What pattern do you see that explains why some sets of numbers make a triangle and some do not? Does it agree with your prediction? Answers Embedded in Student Edition Problems Designing Triangles Experiment: The Side Connection PROBLEM 1.1 Answers 10 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Possible Examples Some lengths can form a triangle. Some lengths won’t allow triangle construction. Side Lengths Triangle Possible? Sketch Different Triangle Possible? 3-4-5 yes no 2-3-4 yes no 4-5-7 yes no 1-1-3 no no 2-4-7 no no 2-3-8 no no Student responses to this question will vary. Possible answers: • If the sum of two lengths is less than the third length, no triangle can be constructed. The two lengths will not “meet” to form a triangle. • Students might discuss that if the sum of the two smaller side lengths equals the third length, a line segment is formed. You need a little more length to “lift up” the sides so angles of the triangle can be formed. • If a set of three numbers makes a triangle, is it possible to make another triangle with a different shape? Explain why. No. A given set of side lengths can be used to make only one triangle. Only one triangle is possible from any three numbers. The orientation of the triangle may change, but it would still be the same triangle. The 3-4-5 is the same triangle as the 4-5-3, but it may look turned. The 3-4 sides are still sharing a vertex. The 4-5 sides are still sharing a vertex. And the 5-3 sides are still sharing a vertex. So it is the same triangle. Note: The adjacent side lengths will stay the same. The term adjacent is introduced later, but you can introduce it now as students explain the “side lengths next to each other.” 1.1 Answers Problem 1.1 Designing Triangles Experiment: The Side Connection 11 SAMPLE

• Will your pattern work on the following triangles? Explain why. An equilateral triangle An isosceles triangle A right triangle Since all of the side lengths are equal, two of the lengths will always add up to more than the third side. With all the sides the same length, only one unique triangle can be made. With two of the side lengths equal, any two of the lengths will always add up to more than the third side. Only one unique triangle can be made because each side touches the other two sides, so there is no way to change the order of the sides to make a different triangle. The two shorter side lengths (legs) add up to more than the third side (hypotenuse). Only one unique triangle can be made because each side touches the other two sides, so there is no way to change the order of the sides to make a different triangle. WHAT IF . . . ? Situation A. Building a Tent Geraldo is building a tent. He has • four 3-foot poles; • two 5-foot poles; • one 6-foot pole; and • one 7-foot pole. He wants to make a triangular-shaped doorframe. 1. How many ways can he make a triangular doorframe? He can make one of each of these triangles for the doorframe: 3-3-3, 3-5-6, 3-5-7, 3-6-7, 5-5-6, 5-5-7, 5-6-7. He can make two 3-3-5 triangles. 2. Which triangle would you use to make the doorframe? Explain your reasoning. A likely solution is to make 3-3-5 frames so there can be two same-size doors. Note that Geraldo can try to use 3-3-6 and 3-3-7, but they will not form a triangle because 3 + 3 = 6 and 3 + 3 < 7. To make a triangle from three given lengths, the sum of the two shortest sides must be greater than the length of the longest side. 5 ft 3 ft 6 ft 7 ft 1.1 Answers 12 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

NOW WHAT DO YOU KNOW? What combinations of three side lengths can be used to make a triangle? How many different triangles are possible for each combination? To make a triangle from three given lengths, the sum of the two shortest sides must be greater than the length of the longest side. Only one triangle is possible with given lengths of the three sides. If sides A and B are the shorter sides and side C is the longest side, then A + B > C. 1.1 Answers Problem 1.1 Designing Triangles Experiment: The Side Connection 13 SAMPLE

Polystrips Name Date Class Note: A polystrip set contains six strips of each length. TEMPLATE LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 14 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Name Date Class Building Triangles LEARNING AID 1.1 Side Lengths Triangle Possible? Sketch Different Triangle Possible? YES or NO YES or NO YES or NO YES or NO YES or NO YES or NO © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 1.1 Designing Triangles Experiment: The Side Connection 15 SAMPLE

Arc of Learning Exploration Introduction Exploration PROBLEM 1.2 Designing Quadrilaterals Experiment: The Side Connection At a Glance The objective of this problem is to extend the Triangle Inequality Theorem into a somewhat parallel result for quadrilaterals. The Initial Challenge will have students building quadrilaterals with different side lengths. The What If . . . ? situations will look at conjectures, reveal the fact that quadrilaterals are not rigid figures, and investigate other polygons. NOW WHAT DO YOU KNOW? Describe what you learned from experiments in building triangles and quadrilaterals. How are different polygons similar and different? How do the differences explain the frequent use of triangles when building structures? Key Terms Materials For each student • Learning Aid 1.2: Building Quadrilaterals For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips (optional) Pacing 1 day Groups 2–3 students A 7–12 C 17–18 E 22 Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Discuss that the results of the sum of two side lengths is greater than the third side length for triangles and the uniqueness of the side-side-side condition for triangles that students found in Problem 1.1. PRESENTING THE CHALLENGE Suggested Question • Can you make a quadrilateral using any four lengths for the sides? If so, is the shape unique? Tell students they are going to conduct an experiment to gather data to help answer this question. They will look for relationships that will let them predict whether four line segments will make a quadrilateral without building it. Explain to students that keeping an accurate record of their data is very important because it allows them to recreate examples as evidence of what they discovered. Make Predictions 16 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS As students work on the Initial Challenge, make sure students keep the question “Can you make a quadrilateral using any four lengths for the sides?” in mind. Encourage them to consider different arrangements of side lengths and to record differences in the shapes that may occur. Also, encourage students to make sketches of their quadrilaterals. For What If . . . ? Situation A, you may want to have half the room investigate pentagons and half the room investigate hexagons. Some groups may want to investigate other polygons as well. PLANNING FOR THE SUMMARY As students are working on making quadrilaterals, look for examples that will make a quadrilateral and ones that do not. Use these examples in the summary. Listen for the way students are talking about the side lengths necessary to create a quadrilateral. SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Each part of this problem focuses on an important property of polygons, so it will be important to review student answers. You can start with a gallery walk for the Initial Challenge. Students should be able to justify answers to the questions: • Can any four lengths form a quadrilateral? • What is unique about the four side lengths that can be used to build squares? Rectangles? Parallelograms? MAKING THE MATHEMATICS EXPLICIT A summary of strategies for finding different quadrilaterals from a given set of side lengths should come from the discussion. Two powerful strategies that focus on different aspects of what determines a quadrilateral are the following: • Put the set of lengths together in different orders. (This technique highlights the role of side lengths in determining a shape.) • Build a quadrilateral from polystrips. Alter its shape by pressing on the sides or vertices of the quadrilateral. A quadrilateral with any given side lengths can form an infinite number of different quadrilaterals. (This technique highlights the role of angles in determining a shape and the lack of rigidity for quadrilaterals.) If four side lengths make a quadrilateral, the shape is not unique. Ask for examples of this, and put them on the board or poster paper. As the discussion continues, add other examples with descriptions. Ask students to explain and record in their notes a clear answer to the build-and-stress activity questions. Ask whether a quadrilateral is rigid. Finally, students should compare the relationships among side lengths of triangles with those of quadrilaterals. You may want to make a poster to show these findings. Include information about angle sums as well. As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Gallery Walk Language 1.2 Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection 17 SAMPLE

EXTENDED LAUNCH—EXPLORE—SUMMARIZE 18 Investigation 1 Designing Triangles Experiment: The Side Connection Problem Overview The objective of this problem is to extend the Triangle Inequality Theorem into a somewhat parallel result for quadrilaterals. The sum of any three side lengths of a quadrilateral is greater than the fourth side. If a quadrilateral can be built from four side lengths, different shapes of quadrilaterals can also be built from those four lengths. Also, the problem reveals the fact that quadrilaterals are not rigid figures. A quadrilateral’s shape will change in response to pressure on the vertex. This is why, when quadrilaterals are used in construction, it is common to see bracing, turning quadrilateral frames into linked triangles. Launch (Getting Started) Connecting to Prior Knowledge Discuss the results of the sum of two side lengths is greater than the third side length for triangles. Remind students of the uniqueness of the side-side-side condition for triangles that students found in Problem 1.1. Presenting the Challenge When the students understand how to manipulate the polystrips, raise the question of the relationship among the lengths of the sides of a quadrilateral. Suggested Question • Can you make a quadrilateral using any four lengths for the sides? If so, is the shape unique? (Students may have some initial thoughts on this. Let one or two students share their opinions.) Tell students they are going to conduct an experiment to gather data to help answer this question. They will look for relationships that will let them predict whether four line segments will make a quadrilateral without building it. Go over the directions for the Initial Challenge. Students are to choose four numbers to be the lengths of the sides of a quadrilateral and then use polystrips to test the lengths to see if they make a quadrilateral. Remind them to record exactly what their numbers are and whether or not they will make a quadrilateral. Then, repeat the test with four new numbers. Have students keep in mind that as they select lengths for the sides, they should try to create interesting and different quadrilaterals. SAMPLE

Distribute Learning Aid 1.2: Building Quadrilaterals. Students can use polystrips or paper polystrips from the Learning Aid Template: Polystrips. Even though students are allowed to work in pairs or small groups, each student should complete their own table. Explain to students that keeping an accurate record of their data is very important because it allows them to re-create examples as evidence of what they discover. (Portrayal) Explore (Digging In) Providing for Individual Needs As students work on the Initial Challenge, make sure students keep the question “Can you make a quadrilateral using any four lengths for the sides?” in mind. Encourage them to consider different arrangements of side lengths and to record differences in the shapes that may occur. Also, encourage students to make sketches of their quadrilaterals. Be sure they are recording the side lengths in the table. You may want to provide extra polystrips so that students can keep various versions of quadrilaterals with the same side lengths to compare. This way they can check to see if the quadrilaterals are the same or different before disassembling them. (Problem-Solving Environment) Suggested Questions • What examples have you created? (Answers will vary.) • What combinations of sides can be used to make quadrilaterals? (Answers will vary.) • What is a summary statement that you could use to help someone who is not here today know how to judge whether four lengths will make a quadrilateral without actually building the quadrilateral? (Explanations will vary.) • How can you represent your conjecture symbolically? (Accept all reasonable answers.) The results from the Initial Challenge can be put on poster paper to share in a gallery walk during the summary. (Portrayal) If a group is not making progress on the question of constructing more than one quadrilateral with a given set of four lengths, share something such as the following to challenge them. (Agency, Identity, Ownership) • In one classroom, a group said they thought they could make more than one quadrilateral with the lengths 6, 8, 10, and 12. They said, “When we put the 10 between the 6 and the 8, the quadrilateral is different from the one we get when we put the 10 between the Extended Launch—Explore—Summarize 19 LES SAMPLE

8 and the 12.” Do you agree or disagree with this group? (This strategy does produce two different quadrilaterals. Putting the 10 between the 6 and the 12 also produces another quadrilateral.) For What If . . . ? Situation A, you may want to have half the room investigate pentagons and half the room investigate hexagons. Some groups may want to investigate other polygons as well. While students are testing pentagons and hexagons, have the groups build examples supporting their conclusions. Make sure to leave plenty of time to discuss their conclusions as a class. • Does anyone have a set of side lengths to try to see if they create a quadrilateral? (Answers will vary.) (Agency, Identity, Ownership) Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? What do we know about side-length relationships among the sides of a polygon? (As students are working on making quadrilaterals, look for examples that will make a quadrilateral and ones that do not. Use these in the summary. Listen for the way students are talking about the side lengths necessary to create a quadrilateral.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Each part of this problem focuses on an important property of polygons, so it will be important to review student answers. You can start with a gallery walk for the Initial Challenge. Students should be able to justify answers. Making the Mathematics Explicit Asking students to construct polystrips figures that demonstrate their thinking will be useful in making the general mathematical conclusions clear and convincing. Have the groups report their findings and share their examples supporting their conclusions. You want them to leave the experience being able to explain what happened and able to design a quadrilateral from a set of side lengths as evidence supporting their findings. A summary of strategies for finding different quadrilaterals from a given set of side lengths should come from the discussion. LES 20 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Two powerful strategies that focus on different aspects of what determines a quadrilateral are the following: 1. Put the set of lengths together in different orders. (This technique highlights the role of side lengths in determining a shape.) 2. Build a quadrilateral from polystrips. Alter its shape by pressing on the sides or vertices of the quadrilateral. A quadrilateral with any given side lengths can form an infinite number of different quadrilaterals. (This technique highlights the role of angles in determining a shape and the lack of rigidity for quadrilaterals.) If four side lengths make a quadrilateral, the shape is not unique. Ask for examples of this, and put them on the board or poster paper. As the discussion continues, add other examples with descriptions. (Portrayal) Students should be able to determine that a parallelogram is formed if opposite side lengths are equal. The parallelogram will become a rectangle as the vertices are pushed to form right angles. Pushing farther will change the rectangle to a nonrectangular parallelogram. Finally, students should compare the relationships among side lengths of triangles with those of quadrilaterals. You may want to make a poster to show these findings. Include information about angle sums as well. Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES Extended Launch—Explore—Summarize 21 SAMPLE

INITIAL CHALLENGE The best way to discover what is so special about quadrilaterals in construction is to build several models and explore the relationship among the side lengths. Use polystrips to make and study several quadrilaterals. Make a Prediction • Will any four side lengths make a quadrilateral? Student predictions will vary based on their past experiences. Conduct the Experiment Equipment › polystrips › fasteners Directions › Choose any four numbers between 2 and 20 for side lengths. Use them as side lengths to build a quadrilateral. › If you can build a quadrilateral, try to build a different quadrilateral with the same side lengths. › Record the data in a table similar to the one shown here. › Make and study several other quadrilaterals. Side Lengths Quadrilateral Possible? Sketch Different Quadrilateral Possible? Answers Embedded in Student Edition Problems Designing Quadrilaterals Experiment: The Side Connection PROBLEM 1.2 Answers 22 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Analyze the Data • What pattern do you see that explains why some sets of numbers make a quadrilateral and some do not? Does this agree with your prediction? To construct a quadrilateral from four given side lengths, the sum of the three shortest side lengths must be greater than the longest side length. When it is possible to make one quadrilateral from four side lengths, it is possible to make many different shapes from the same side lengths. Note that when you go from a 3-5-3-5 to a 3-3-5-5, you will have a different combination of sides that share a vertex. This forms a different shape. • What combination of side lengths is needed to build squares? Rectangles? Parallelograms? Special quadrilaterals can be made with the following combinations of side lengths. • All four side lengths are equivalent: rhombus and square (if the angles are 90°). • Three side lengths are equivalent: a special type of isosceles trapezoid. • Two side lengths are equivalent, or one pair of side lengths is equivalent: isosceles trapezoid (if the opposite sides are equal length). • Two pairs of side lengths are equivalent: parallelograms (if the opposites sides are equal length), rectangles (if the opposites sides are equal length and angles are 90°), and kites (if adjacent sides are equal length). 1.2 Answers Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection 23 For example, a 3-5-3-5 would form a parallelogram. If you reorder the sides to 3-3-5-5, you will have a kite. SAMPLE

• Test your conjectures on the following side lengths. Which ones form a quadrilateral? Are any of these squares? Rectangles? Parallelograms? Explain why. Shape 1 6, 10, 15, 15 Shape 2 3, 5, 10, 20 Shape 3 8, 8, 10, 10 Shape 4 12, 20, 6, 9 Quadrilateral? Yes 6 + 10 + 15 > 15 Quadrilateral? No 3 + 5 + 10 < 20 Quadrilateral? Yes 8 + 8 + 10 > 10 Quadrilateral? Yes 6 + 9 + 12 > 20 Special Quadrilateral: Isosceles trapezoid (if opposites sides are same length) 15 15 10 6 Special Quadrilateral: Parallelogram (if the opposites sides are equal length) 8 8 10 10 Rectangle (if the opposites sides are equal length and angles are 90°) 10 10 8 8 Kite (if adjacent sides are equal length) 1010 88 Special Quadrilateral: Trapezoid (if the opposites sides are parallel) 6 12 20 9 None of the given side-length combinations can be used to form a square because a square requires all side lengths to be equal and the angles to be 90°. 1.2 Answers 24 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

WHAT IF . . . ? Situation A. Testing More Polygons Michal wondered if there was a side-length relationship for other polygons. She used polystrips to build pentagons and hexagons. 1. What are some side lengths that make pentagons? Hexagons? Answers will vary. Some examples include: Pentagons: 5-5-5-5-5, 5-4-3-2-1 Hexagons: 7-7-7-7-7-7-7, 8-4-4-3-3-2 2. What relationship among side lengths could she observe? The longest side cannot be larger than the sum of the other side lengths. If the longest side is too long, the other side lengths will “flatten out” and not have any angle to create a shape. For example, with fives sides that measure 20-3-3-3-3, the longest side is too long. The other lengths will “flatten out.” For example, with fives sides that measure 10-3-3-3-3, the longest side is not too long. The other four lengths total 12 units. There is “room for an angle other that 0° to form.” 3. What is the largest polygon your group can make with your set of polystrips? How many sides does it have? Answers will vary based on how many polystrips the group has. With n number of polystrips, they can make an n-gon. 4. Look back at the angle sizes of the polygons you have made. What do you notice about the measure of the angles as the number of sides of a polygon increases? Answers will vary based on the polygons that the students make. Some students may not notice any patterns at this time. If students make equilateral shapes, they may notice the angle measures getting larger with the increase in the number of sides. Note: Students will have conjectures at this time that can be investigated in later problems. This question foreshadows the patterns that students will examine in Investigation 3. 1.2 Answers Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection 25 SAMPLE

Situation B. Beatrice Worries About Rigidity Beatrice’s Experiment I wanted to know how the rigidity of triangles and quadrilaterals reacts to stress. I thought of the force applied to a roof with heavy snow or the force applied to a bridge by a car or train. So, I designed an experiment. I chose three different lengths that I knew would create a triangle. And I chose four different lengths that I knew would create a quadrilateral. I wanted to know what would happen when I apply force to a side or vertex. Push Down Push Down 1. What do you think happened when Beatrice pushed down on the vertex of a triangle? On a side or vertex of a quadrilateral? Triangle When you push down on a vertex as suggested, the triangle will hold firm or rigid until the sides themselves buckle. Quadrilateral When you push down on a vertex as suggested, the quadrilateral will quickly deform into different shapes (unless the connections at the vertices are very tight). 2. What would happen if she repeated this test on a pentagon or hexagon? Pentagon or Hexagon When you push down on a vertex as suggested, the pentagon or hexagon will quickly deform into different shapes (unless the connections at the vertices are very tight). The stress test explains why triangles are used to hold structural shapes firm. NOW WHAT DO YOU KNOW? What do we know about side-length relationships among the sides of a polygon? In building triangles, quadrilaterals, and other polygons, each side must be less than the sum of the others. For example, if sides A, B, and C are the shorter sides and side D is the longest side, then A + B + C > D. Only one triangle can be formed when the side lengths meet to form a triangle. Unlike triangles, if a quadrilateral can be formed with four side lengths, then there are many different quadrilaterals that can be constructed. For example, with side lengths 3, 4, 3, 4 as adjacent sides, I get a parallelogram. If I change the order to side lengths 3, 3, 4, 4 as adjacent sides, I get a kite. 1.2 Answers 26 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Name Date Class Note: A polystrip set contains six strips of each length. Polystrips TEMPLATE LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection 27 SAMPLE

Name Date Class Side Lengths Quadrilateral Possible? Sketch Different Quadrilateral Possible? YES or NO YES or NO YES or NO YES or NO YES or NO YES or NO Building Quadrilaterals LEARNING AID 1.2 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 28 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Arc of Learning Introduction Exploration Introduction Exploration NOW WHAT DO YOU KNOW? What do you know about the rigidity of shapes, and how does this explain the frequent use of triangles in building structures? Key Terms Materials For each student • Learning Aid 1.3A: Diagonals and Rigidity • Learning Aid 1.3B: Diagonals and Rigidity Shapes (optional) For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips Pacing 1 day Groups 2 students A 13–14 C 19–20 E 23 Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Revisit What If . . . ? Situation B from Problem 1.2 on Beatrice’s experiment with rigidity of triangles and quadrilaterals. PRESENTING THE CHALLENGE Discuss the introduction of Beatrice’s wondering about adding diagonals to make polygons more rigid. Suggested Questions • What does it mean to be rigid? • Are triangles rigid? Quadrilaterals? • What do you think would happen if you added a diagonal to one of the quadrilaterals you made yesterday? Explain the experiment in the Initial Challenge. • If a polygon with n sides is not rigid, what is the minimum number of diagonals needed to make it rigid? Making Predictions PROBLEM 1.3 At a Glance In this problem, students will investigate shapes that aren’t very rigid and how to increase the rigidity with diagonals. Students will conduct an experiment in the Initial Challenge to find that the number of diagonals needed to make a polygon rigid is 3 less than the number of sides. Students will generalize in words and symbols the relationship between the number of diagonals that make it rigid and the created embedded triangles in the polygons. Problem 1.3 Rigidity Experiment 29 Rigidity Experiment SAMPLE

1.3 30 Investigation 1 Designing Triangles Experiment: The Side Connection Facilitating Discourse (continued) Teacher Moves (continued) EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Circulate to see that students are building the polygons with diagonals correctly. Suggested Questions • How do you know that you have built a _________ (insert name of polygon)? • What tells you the shape is rigid? • Do you see a pattern in the data in your table? • Does your sketch give you any ideas on how many diagonals are needed? PLANNING FOR THE SUMMARY As you are circulating during the Explore, select student work to use in the summary that will push the conversation on minimum number of diagonals needed to make a polygon rigid. Select polygons that students have made of various side lengths to use as examples. Listen for students who are finding the embedded triangles in the polygons as they add the diagonals Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students share their polygons with diagonals that they made with polystrips. Have students share the conclusion they found for the minimum number of diagonals needed to make an n-gon rigid. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • Do you agree with _________’s polygon and the number of diagonals they have chosen? • What did you notice in your sketches of the polygons and the minimum number of diagonals? • What is the minimum number of diagonals for an n-gon? • Did this generalization work for a heptagon, octagon, and/or nonagon? • Do you agree with Beatrice? • Does Beatrice’s observation about creating embedded triangles when adding the diagonals appear in the polygons you made with the polystrips? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking SAMPLE

EXTENDED LAUNCH—EXPLORE—SUMMARIZE Problem Overview In this problem, students will investigate shapes that aren’t rigid and explore how to increase the rigidity with diagonals. Students will conduct an experiment to find that the number of diagonals needed to make a polygon rigid is 3 less than the number of sides. Launch (Getting Started) Connecting to Prior Knowledge Revisit What If . . . ? Situation B from Problem 1.2 on Beatrice’s experiment with rigidity of triangles and quadrilaterals. Presenting the Challenge Discuss the introduction of Beatrice’s wondering about adding diagonals to make polygons more rigid. Suggested Questions • What does it mean to be rigid? (Unable to bend or be forced out of shape; not flexible. Accept all reasonable answers.) • Are triangles rigid? (Yes.) Quadrilaterals? (No.) • Predict what you think would happen if you added a diagonal to one of the quadrilaterals you made yesterday? (Answers will vary.) Note: This idea is explored in the problem. Explain the experiment in the Initial Challenge. • If a polygon with n sides is not rigid, what is the minimum number of diagonals needed to make it rigid? (Answers will vary.) Distribute Learning Aid 1.3A: Diagonals and Rigidity and polystrips. Have students work with a partner on this problem. Distribute Learning Aid 1.3B: Diagonals and Rigidity Shapes if you feel students need support drawing the shapes. Explore (Digging In) Providing for Individual Needs Circulate to see that students are building the polygons with diagonals correctly. Suggested Questions • How do you know that you have built a _________ (insert name of polygon)? (Answers will vary. Make sure students are connecting the names with the correct number of sides.) Extended Launch—Explore—Summarize 31 SAMPLE

• What tells you the shape is rigid? (Answers will vary. Students should be able to talk about not being able to smash it when pushing on a vertex.) • Do you see a pattern in the data in your table? (Answers will vary. Help focus on the number of sides and diagonals.) • Does your sketch give you any ideas on how many diagonals are needed? (Answers will vary.) Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? What do you know about the rigidity of shapes, and how does this explain the frequent use of triangles in building structures? (As you are circulating during the Explore, select student work to use in the summary that will push the conversation on minimum number of diagonals needed to make a polygon rigid. Select polygons that students have made of various side lengths to use as examples. Listen for students who are finding the embedded triangles in the polygons as they add the diagonals.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Discuss how the addition of a diagonal to a quadrilateral produces a rigid figure. Ask students why the diagonal produces a rigid figure. Some students may bring up that it has something to do with the two triangles that are formed. Note: When a diagonal strip is placed in the quadrilateral, it becomes rigid. That is, if the strip stays in the same plane, then the shape is unique. If we are allowed to flip two adjoining strips through a third dimension, we would be able to form a different shape. Have students share their polygons with the diagonals that they made with polystrips. Have students share the conclusion they found for the minimum number of diagonals needed to make an n-gon rigid. Making the Mathematics Explicit Suggested Questions • Do you agree with _________’s polygon and the number of diagonals they have chosen? (Answers will vary. Repeat for several polygons and with several students sharing their polygon.) 32 Investigation 1 Designing Triangles Experiment: The Side Connection LES SAMPLE

• What did you notice in your sketches of the polygons and the minimum number of diagonals? (Answers will vary. The number of diagonals is 3 less than the number of sides of the polygon.) • What is the minimum number of diagonals for an n-gon? (Number of sides of the polygon minus 3.) • Did this generalization work for a heptagon, octagon, and/or nonagon? (Yes. Have students show this with their polystrips.) • Do you agree with Beatrice? (Yes.) • Does Beatrice’s observation about creating embedded triangles when adding the diagonals appear in the polygons you made with the polystrips? (Yes. Have students demonstrate this with their polygons.) Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. See answers to the problem for more information about the strategies and embedded mathematics. REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES Extended Launch—Explore—Summarize 33 SAMPLE

INITIAL CHALLENGE When Beatrice was building polygons in Problem 1.2, she noticed that if she added one diagonal to a quadrilateral, the shape was rigid. Recall that a diagonal is a line segment that joins two nonadjacent vertices of a polygon. She wondered if adding diagonals to other polygons that were not rigid would make them rigid. Make a Prediction • If a polygon with n sides is not rigid, what is the minimum number of diagonals needed to make it rigid? Student predictions will vary based on their past experiences. Conduct the Experiment Equipment › polystrips › fasteners Directions › Build a triangle, quadrilateral, pentagon, and hexagon with your polystrips. What is the least number of diagonals needed to make it rigid? Record the data in a table. › Repeat by building a different quadrilateral, pentagon, and hexagon. Record the data in a table. Shape The Minimum Number of Diagonals Make a Sketch triangle quadrilateral pentagon hexagon Answers Embedded in Student Edition Problems Rigidity Experiment PROBLEM 1.3 Answers 34 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Shape The Minimum Number of Diagonals Make a Sketch heptagon octagon nonagon n-gon Analyze the Data Shape The Minimum Number of Diagonals Make a Sketch triangle 0 quadrilateral 1 pentagon 2 hexagon 3 heptagon 4 octagon 5 nonagon 6 n-gon number of sides − 3 n − 3 1.3 Answers Problem 1.3 Rigidity Experiment 35 SAMPLE

• Was your prediction correct? That is, what is the minimum number of diagonals needed to make a polygon rigid? Answers about predictions will vary. To compute the number of diagonals for each shape, subtract 3 from the number of sides. • If the pattern continues, fill in the table with the minimum number of diagonals needed for a heptagon, octagon, nonagon, and n-gon. See previous chart. • Test your answer by building one of these polygons and adding the minimum number of diagonals. Does it work? If you make the shape with polystrips and support it with n − 3 diagonals, it will be rigid. WHAT IF . . . ? Situation A. Beatrice Again Beatrice’s Observation I noticed that when I add the minimum number of diagonals to a shape, I am creating several embedded triangles. I think this has something to do with why the shape is now rigid. Do you agree with Beatrice? Explain why. Yes. Triangles are rigid shapes. And putting in the minimum number of diagonals subdivides (or decomposes) the shape into triangle-shaped regions. This adds the strength/rigidity of triangles. Note: Many students may make this same observation during the Initial Challenge. NOW WHAT DO YOU KNOW? What do you know about the rigidity of shapes, and how does this explain the frequent use of triangles in building structures? While triangles are rigid figures, quadrilaterals (and all other polygons with more than three sides) are not. This explains the use of triangles in building structures. 1.3 Answers 36 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE

Name Date Class Note: A polystrip set contains six strips of each length. Polystrips LEARNING AID TEMPLATE © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 1.3 Rigidity Experiment 37 SAMPLE

Name Date Class Shape The Minimum Number of Diagonals Make a Sketch Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon n-gon Diagonals and Rigidity 1.3A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 38 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE