Mathematics
Grade 10
15 min
Find the next shape in a repeating pattern
Find the next shape in a repeating pattern
What you'll learn
- Identify the amplitude, period, phase shift, and vertical shift of a given sine function from its equation in the form y = A sin(B(x - C)) + D with 80% accuracy on a written quiz.
- Determine the domain and range of sine functions, including transformations, and express them using interval notation with 75% accuracy on a problem set.
- Graph sine functions and their transformations (amplitude, period, phase shift, and vertical shift) accurately on graph paper and using graphing software, demonstrating at least 80% accuracy based on a rubric.
- Explain how changes in the parameters A, B, C, and D in the equation y = A sin(B(x - C)) + D affect the graph of the sine function, providing justifications based on function transformations with supporting examples in a short essay.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the core repeating unit in a geometric pattern of congruent triangles.
Describe the specific sequence of rigid transformations (translation, rotation, reflection) that generates a pattern.
Use congruence postulates (SSS, SAS, ASA) to justify that shapes within a pattern are congruent.
Apply algebraic rules for transformations on a coordinate plane to predict the vertices of the next triangle in a sequence.
Determine the position and orientation of the nth triangle in a repeating pattern.
Justify their prediction for the next shape using the precise mathematical language of geometry and transformations.
Ever noticed the beautiful, repeating patterns on a tile floor or in a kaleidoscope? 💠 How do artists and designers create those perfect, endless...
2
Key Concepts & Vocabulary
TermDefinitionExample
Congruent TrianglesTriangles that have the exact same size and shape. All pairs of corresponding sides and corresponding angles are equal.If ΔABC ≅ ΔXYZ, it means AB = XY, BC = YZ, AC = XZ, and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z.
Rigid Transformation (Isometry)A transformation of the plane that preserves distance and angle measure. The original figure (pre-image) and the transformed figure (image) are congruent.Translations, rotations, and reflections are the three main rigid transformations.
TranslationA rigid transformation that 'slides' every point of a figure the same distance in the same direction along a vector.Translating ΔABC with vertices A(1,2), B(3,1), C(2,4) by the vector <4, -1> results in ΔA'B'C' with vertices A'(5,1), B'...
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Core Formulas
Translation Rule
T_{a,b}(x, y) = (x + a, y + b)
Use this to find the coordinates of a point after sliding it 'a' units horizontally and 'b' units vertically. If 'a' is negative, the slide is to the left; if 'b' is negative, the slide is down.
Rotation Rules (about the origin)
90° counter-clockwise: R_{90°}(x, y) = (-y, x) \\ 180°: R_{180°}(x, y) = (-x, -y) \\ 270° counter-clockwise: R_{270°}(x, y) = (y, -x)
Use these rules to find the coordinates of a point after rotating it around the origin (0,0). For clockwise rotations, use the equivalent counter-clockwise rotation (e.g., 90° clockwise is 270° counter-clockwise).
Reflection Rules
Across x-axis: r_{x-axis}(x, y) = (x, -y) \\ Across y-axis: r_{y-axis}(x, y) = (-x, y) \\ Across...
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Challenging
A pattern is defined by the recursive rule: T_{n+1} is generated by applying a 180° rotation about the origin to T_n, followed by a translation T_{-2, 3}. If T1 has vertex A(4, 0), what are the coordinates of the corresponding vertex A''' on T4?
A.(4, 0)
B.(-4, 6)
C.(6, -3)
D.(-8, 9)
Challenging
A transformation maps triangle vertices (x, y) to (-y, -x). Which of the following describes this non-standard transformation?
A.270° counter-clockwise rotation about the origin.
B.reflection across the x-axis followed by a reflection across the y-axis.
C.reflection across the line y = -x.
D.180° rotation followed by a reflection across the line y=x.
Challenging
A pattern is generated by a glide reflection. Triangle 1 with vertex K(3, 2) is mapped to Triangle 2 with corresponding vertex K'( -3, 5). The transformation is a reflection across the y-axis followed by a translation. What is the translation vector?
A.<0, 3>
B.< -6, 3 >
C.<0, 7>
D.< -6, 0 >
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Find the next shape in a repeating pattern is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Find the next shape in a repeating pattern?
You'll be able to: Identify the amplitude, period, phase shift, and vertical shift of a given sine function from its equation in the form y = A sin(B(x - C)) + D with 80% accuracy on a written quiz; Determine the domain and range of sine….
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This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.