This text resource has associated with it the Geometry (G) Domain of the Common Core Mathematics Standards.
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In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Numbers
|
1 - Represent (whole numbers with sets of objects)
2 - Compare (whole numbers with sets of objects)
2 - Describe (shapes and space)
|
1 - I can represent whole numbers with sets of objects.
4 - I can compare sets of objects using whole numbers.
2 - I can describe shapes and space.
|
A. Sets of objects can be represented by numbers.
B. Objects can have different shapes and occupy different spaces.
|
A.1 How can sets of objects be represented?
B.1 How can an object's shape or the space it occupies be described?
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Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
1 - Identify (Shapes)
2 - Describe (Shapes)
|
1 - I can identify mathematical shapes.
2 - I can describe mathematical shapes.
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A. There are different mathematical shapes each with their unique attributes.
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A.1 What are mathematical shapes? What are the differences between them?
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Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Objects in Environment
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2 - Describe (Objects by name of their shapes)
2 - Describe (Objects relative positions, above, below, beside, in front of, behind, next to)
|
2 - I can describe objects by their shapes.
2 - I can describe relative positions of objects.
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A. Objects can be described by their shapes.
B. Objects can be described by their positions.
|
A.1 How can shapes be used to describe objects?
B.1 How can positions of objects describe them?
|
Correctly name shapes regardless of their orientations or overall size.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Name (Shapes regardless of orientation and size)
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2 - I can name shapes regardless of orientation or size.
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A. Objects with the same shape have the same name regardless of orientation or size.
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A.1 How does orientation or size of an object affect its shape?
|
Identify shapes as two-dimensional (lying in a plane, “flat”) or three- dimensional (“solid”).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-dimensional Shapes
Three-dimensional Shapes
|
1 - Identify (two-dimensional shapes, flat plane)
1 - Identify (three-dimensional shapes, solids)
|
1 - I can identify two dimensional shapes.
1 - I can identify three-dimensional shapes.
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A. Shapes can be identified by their number of dimensions.
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A.1 What is the difference between two- and three-dimensional shapes?
|
Analyze, compare, create, and compose shapes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Untitled
|
2 - Compare (Shapes)
4 - Analyze (Shapes)
6 - Create (Shapes)
6 - Compose (Shapes)
|
2 - I can compare shapes.
4 - I can analyze shapes.
5 - I can create shapes.
6 - I can compose shapes.
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A. Shapes can be created, compared, and analyzed.
|
A.1 What mathematical shapes can be created? How do I create them?
A.2 How can I analyze a shape?
A.3 How can I compare two or more shapes?
|
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shape Attributes
|
2 - Describe (Two- Dimensional Shapes)
2 - Describe (Three- Dimensional Shapes)
4 - Compare (Two- Dimensional Shapes)
4 - Compare (Three- Dimensional Shapes)
4 - Analyze (Two- Dimensional Shapes)
4 - Analyze (Three- Dimensional Shapes)
|
4 - I can compare two -dimensional shapes by their attributes.
4 - I can compare three -dimensional shapes by their attributes.
4 - I can analyze two -dimensional shapes by their attributes.
4 - I can analyze three -dimensional shapes by their attributes.
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A. Shapes may have identical attributes while having other attributes that are different.
|
A.1 What shape attributes must be the same for two objects to be the same type?
A.2 What shape attributes can vary for two objects to be the same type?
|
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
|
3 - Draw (Shapes)
6 - Model (World-based shapes)
6 - Build (Shapes with components)
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3 - I can draw shapes.
6 - I can model real-world shapes using physical components.
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A. Shapes found in the world can be modeled using physical components or through drawings.
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A.1 What shapes can be modeled using physical components?
A.2 How can we benefit from using models?
A.3 What shapes can be modeled using drawings.
|
Compose simple shapes to form larger shapes.
For example, “Can you join these two triangles with full sides touching to make a rectangle?"
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
6 - Compose (Shapes from other shapes)
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6 - I can compose shapes by combining other shapes.
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A. Complex shapes may be constructed from simple shapes.
|
A.1 What shapes can be joined to create other shapes? What are the constructed shapes?
A.2 What simple shapes can I make by breaking apart other shapes?
|
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Number Relationships
Measurement
Shapes
|
2 - Understand (addition and subtraction)
2 - Understand (addition and subtraction strategies within 20)
2 - Understand (whole number relationships and place value, including grouping in tens and ones)
2 - Understand (linear measurement)
2 - Measure (lengths as iterating length units)
4 - Reasoning/Analyze (attributes of geometric shapes)
6 - Compose/Decompose (geometric shapes)
|
2 - I can understand addition and subtraction.
3 - I can understand addition and subtraction strategies within 20.
2 - I can understand whole number relationships and place value, including grouping in tens and ones.
2 - I can understand linear measurement.
2 - I can measure lengths using length units.
2 - I can understand the attributes of geometric shapes.
6 - I can compose geometric shapes.
6 - I can decompose geometric shapes.
|
A. The number of objects can be added or subtracted.
B. The value of a whole number depends on its place value.
C. Linear measurement requires the use of length units.
D. The attributes of geometric shapes allow you to compose or decompose them into other shapes.
|
A.1 What is addition? How do you use it?
A.2 What is subtraction? How do you use it?
B.1 How can the place value of a whole number be found?
C.1 What are length units? How is linear measurement and the measuring of lengths related to length units?
D.1 What are the attributes of geometric shapes? How can they be composed or decomposed?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Reason with shapes and their attributes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
|
4 - Reason (shapes and their attributes)
|
4 - I can reason with shapes and their attributes.
|
A. Shapes have attributes
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A.1 What are shapes? How do their attributes help distinguish them?
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Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Distinguish (between defining attributes and non-defining attributes of shapes)
6 - Build (shapes with defining attributes)
6 - Draw (shapes with defining attributes)
|
1 - I can identify defining attributes of shapes (e.g., triangles are closed and three-sided).
1 - I can identify non-defining attributes of shapes (e.g., color, orientation, overall size).
2 - I can distinguish between defining attributes and non-defining attributes of shapes.
|
A. Shapes have both defining and non-defining attributes.
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A.1 How can one distinguish between the defining and non-defining attributes of a particular shape?
|
Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional Shapes
Three-Dimensional Shapes
Shapes
|
6 - Compose (two-dimensional shapes)
6 - Compose (three-dimensional shapes)
6 - Create (composite shapes from two-dimensional shapes)
6 - Create (composite shapes from three-dimensional shapes)
6 - Compose (new shapes from composite shapes)
|
6 - I can compose two-dimensional shapes such as rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles.
6 - I can compose three-dimensional shapes such as cubes, right rectangular prisms, right circular cones, and right circular cylinders.
6 - I can create composite shapes from two-dimensional shapes.
6 - I can create composite shapes from three-dimensional shapes.
6 - I can compose new shapes from composite shapes.
|
A. Shapes can be composed of other shapes.
|
A.1 How can two-dimensional shapes of different kinds be combined into other shapes?
A.2 How can three-dimensional shapes of different kinds be combined into other shapes?
|
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
Partitions
Share Phrases
|
3 - Partition (circles and rectangles into two and four equal shares)
2 - Describe (shares using the words "halves," "fourths," and "quarters")
2 - Describe (shares using the phrases "half of," "fourth of," and "quarter of")
2 - Describe (the whole as "two of," or "four of" the shares)
2 - Understand (decomposing into more equal shares creates smaller shares)
|
3 - I can partition circles and rectangles into two and four equal shares.
2 - I can describe shares using the words "halves," "fourths," and "quarters."
2 - I can describe shares using the phrases "half of," "fourth of," and "quarter of."
2 - I can describe the whole as "two of," or "four of" the shares.
2 - I can understand how decomposing existing shares into more equal shares creates smaller shares.
|
A. Circles and rectangles can be partitioned into equal shares.
|
A.1 What are equal shares? How can circles and rectangles be partitioned?
A.2 How can the equal parts of partitioned circles and rectangles be described?
A.3 What results when equal shares of a shape are decomposed into more equal parts?
|
In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
3 - Extend/ Understand (base-ten notation)
3 - Build (addition and subtraction fluency)
3 - Use (standard units of measure)
1 - Describe (shapes)
4 - Analyze (shapes)
|
1 - I can understand base-ten notation.
3 - I can demonstrate greater fluency with addition and subtraction.
3 - I can use standard units of measure.
3 - I can describe and analyse shapes.
|
A. The value of a whole number depends on its place value.
B. Addition and subtraction are essential math skills for everyday life.
C. Measuring length requires the use of standard units of length.
D. The attributes of geometric shapes can be described and analyzed.
|
A.1 What is base-ten notation? How is it used?
B.1 How can addition and subtraction be used to solve real world problems?
C.1 What are standard units of measure? How can they be used?
D.1 What are examples of geometric shape attributes? How can they be described and analyzed?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Reason with shapes and their attributes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
|
4 - Reason (shapes and their attributes)
|
4 - I can reason with shapes and their attributes.
|
A. Shapes have attributes
|
A.1 What are shapes? How do their attributes help distinguish them?
|
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shape Names
|
1 - Recognize (shapes having specified attributes)
4 - Draw (shapes having specified attributes)
1 - Identify (triangles, quadrilaterals, pentagons, hexagons, cubes)
|
1 - I can recognize shapes having specified attributes.
4 - I can draw shapes having specified attributes.
1 - I can identify shapes including triangles, quadrilaterals, pentagons, hexagons, and cubes.
|
A. Shapes can be identified by their attributes.
|
A.1 What are the different types of shapes? How can these different shapes be identified?
A.2 How does one draw a triangle, quadrilateral, pentagon, hexagon, and cube?
|
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Rectangle Partition
|
3 - Partition (rectangle into rows and columns of same-size squares)
2 - Count (squares within partitioned rectangle)
|
3 - I can partition a rectangle into rows and columns of same-size squares.
2 - I can count the total number of squares in a partitioned rectangle.
|
A. Same-size squares can be counted from partitioned rectangles.
|
A.1 How can same-size squares be created within a rectangle?
A.2 How can these same-size squares be counted?
|
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
Partitions
Shares as a Whole
|
3 - Partition (circles and rectangles into two, three, or four equal shares)
2 - Describe (shares using the words "halves," "thirds," "half of," and "third of," etc.)
2 - Describe (the whole as "two halves," "three thirds," and "four fourths.")
2 - Recognize (equal shares of identical wholes need not have the same shape)
|
3 - I can partition circles and rectangles into two, three, or four equal shares.
2 - I can describe shares using the words "halves," "thirds," "half of," and "third of," etc.
2 - I can describe the whole as "two halves," "three thirds," and "four fourths."
2 - I can recognize that equal shares of identical wholes need not have the same shape.
|
A. Circles and rectangles can be partitioned into equal shares and described.
|
A.1 What are equal shares? How can circles and rectangles be partitioned?
A.2 How can the equal parts of partitioned circles and rectangles be described?
A.3 What results when equal shares of a shape are decomposed into more equal parts?
|
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Operations
Structures
Shapes
|
2 - Understand (multiplication and division)
2 - Understand (multiplication and division strategies within 100)
2 - Understand (fractions)
2 - Understand (unit fractions, fractions with numerator 1)
2 - Understand (structure of rectangular arrays and area)
1 - Describe (two-dimensional shapes)
4 - Analyze (two-dimensional shapes)
|
1 - 2 - I can understand multiplication and division.
1 - 3 - I can understand multiplication and division strategies within 100.
3 - I can understand fractions, especially unit fractions (fractions with numerator 1).
3 - I can understand the structure of rectangular arrays and of area.
1 - I can describe two-dimensional shapes.
4 - I can analyze two-dimensional shapes.
|
A. The number of objects can be multiplied and divided.
B. The number one over another number in a fraction makes a unit fraction.
C. Area involves height and width.
D. The attributes of two-dimensional shapes can be described and analyzed.
|
A.1 What is multiplication? How do you use it?
A.2 What is division? How do you use it?
B.1 What is a unit fraction? How can it be used?
C.1 What is a rectangular array? What is area? How are rectangular arrays and area related?
D.1 What are examples of two-dimensional shapes? How can they be described and analyzed?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Reason with shapes and their attributes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
|
4 - Reason (shapes and their attributes)
|
4 - I can reason with shapes and their attributes.
|
A. Shapes have attributes
|
A.1 What are shapes? How do their attributes help distinguish them?
|
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shape Categories
|
2 - Understand (different shape categories may share attributes)
2 - Understand (shared attributes can define a larger category)
1 - Recognize (rhombuses, rectangles, and squares are quadrilaterals)
4 - Draw (nonrhombuse, nonrectangle, and nonsquare quadrilaterals)
|
2 - I can understand that different shape categories may share the same attributes.
2 - I can understand that shared attributes can define a larger shape category.
1 - I can recognize rhombuses, rectangles, and squares are quadrilaterals.
4 - I can draw quadrilaterals that are not rhombuses, rectangles, or squares.
|
A. Shared attributes of different geometric shapes can define a larger shape category.
|
A.1 How can different categories of shapes belong to a larger category? What shapes are examples of this?
A.2 How do you draw quadrilaterals like rectangles, squares, and rhombuses?
A.3 How do you draw quadrilaterals that are not rectangles, squares, and rhombuses?
|
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Parts of Shapes
|
3 - Partition (shapes into equal area parts)
2 - Express (shape part as a unit fraction of the whole)
|
3 - I can partition shapes into equal are parts.
2 - I can express the area of each part as a unit fraction of the whole.
|
A. Shapes can be partitioned into equal area parts.
|
A.1 How can shapes be partitioned into equal area parts?
A.2 How can equal area parts of a shape be described using fractions?
|
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Multi-Digit Operations
Fractions
|
3 - Understand (multi-digit multiplication)
3 - Understand (division with multi-digit dividends)
2 - Understand (fraction equivalence)
2 - Understand (addition and subtraction of fractions with like denominators)
2 - Understand (multiplication of fractions by whole numbers)
3 - Understand (geometric figure analysis and classification)
2 - Classify (geometric figures by their properties, parallel sides, perpendicular sides, particular angle measures, symmetry)
4 - Analyze (geometric figures by their properties, parallel sides, perpendicular sides, particular angle measures, symmetry)
|
2 - I can understand and apply (develop fluency) multi-digit multiplication.
3 - I can understand and apply (develop fluency) dividing to find quotients involving multi-digit dividends.
2 - I can understand fraction equivalence.
2 - I can understand addition and subtraction of fractions with like denominators.
2 - I can understand multiplication of fractions by whole numbers.
2 - I can understand geometric figures can be analyzed and classified based on their properties.
2 - I can classify geometric figures based on their properties, such as parallel sides, perpendicular sides, particular angle measures and symmetry.
4 - I can Analyze geometric figures based on their properties, such as parallel sides, perpendicular sides, particular angle measures and symmetry.
|
A. A number of objects can be multiplied and divided using multi-digit numbers.
B. Fractions may look different but still be equal.
C. Math operations can be done on fractions to solve problems.
D. Geometric figures can be classified and analyzed by their properties.
|
A.1 What is multiplication? How do you use it with multi-digit numbers?
A.2 What is division? How do you use it with multi-digit numbers?
B.1 Can two fractions look different but still be equal? Why?
C.1 How can fractions be added?
C.2 How can fractions be subtracted?
C.3 How can fractions be multiplied with whole numbers?
D.4 What are the different properties of geometric figures? How can they be used to classify or analyze them?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Shapes
|
1 - Identify (lines)
1 - Identify (angles)
3 - Draw (lines)
3 - Draw (angles)
2 - Classify (shapes by line and angle properties)
|
1 - I can identify a line.
3 - I can draw a line.
1 - I can identify an angle.
3 - I can draw an angle.
1 - I can classify a shape using its line and angle properties.
|
A. A shape can be classified by its lines and angles.
|
A.1 How can you draw and identify lines and angles?
A.2 How can a shape be classified by its lines and angles?
|
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Properties of two-dimensional figures
Line properties
|
3 - Draw (points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular lines, parallel lines)
1 - Identify (properties of two-dimensional figures)
|
3 - I can draw points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular lines, parallel lines.
1 - I can Identify properties of two-dimensional figures.
|
A. Two-dimensional figures have identifiable properties.
|
A.1 What properties of two-dimensional figures can be identified? How can you describe them?
|
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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional Figures
|
2 - Classify (two-dimensional figures on the presence/absence of parallel or perpendicular lines)
2 - Classify (two-dimensional figures on the presence/absence of angles of a specified size)
1 - Recognize (right triangles as a category)
1 - Identify (right triangles)
|
2 - I can classify two-dimensional figures on the presence or absence of parallel or perpendicular lines.
2 - I can classify two-dimensional figures on the presence or absence of angles of a specified size.
1 - I can recognize right triangles as a category.
1 - I can identify right triangles.
|
A. Right triangles have the property of two perpendicular lines.
|
A.1 What is the difference between parallel and perpendicular? How can this be demonstrated in a drawing?
A.2 What is the key property of a right triangle? How do you draw a right triangle?
|
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional figure
|
1 - Recognize (line of symmetry, folded on line into matching parts)
1 - Identify (line-symmetric figures)
3 - Draw (lines of symmetry)
|
1 - I can recognize that when a two-dimensional figure is folded along a line of symmetry it forms two matching parts.
1 - I can identify line-symmetric figures.
3 - I can draw lines of symmetry.
|
A. Some figures can be folded into two matching parts.
|
A.1 How can figures be folded into two matching parts?
|
In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Fractions
Decimals
Concept
|
3 - Develop fluency/Apply (addition, subtraction of fractions)
2 - Understand (multiplication of fractions)
2 - Understand (division of fractions, unit fractions divided by whole numbers, whole numbers divided by unit fractions)
3 - Extend/Apply (division to 2-digit divisors)
4 - Integrate (decimal fractions, place value system)
2 - Understand (decimal operations to hundredths)
3 - Develop fluency/Apply (whole number and decimal operations)
2 - Understand (volume)
|
3 - I can apply (develop fluency) addition and subtraction of fractions.
2 - I can understand the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions).
3 - I can apply (extend) division to 2-digit divisors.
4 - I can integrate decimal fractions into the place value system.
2 - I can understand operations with decimals to hundredths.
3 - I can apply (develop fluency) whole number and decimal operations.
2 - I can understand volume.
|
A. Fraction problems can be solved using all four operations; addition, subtraction, multiplication, and division.
B. Fractions can be expressed as decimals.
C. The location of the decimal in a number determines its value.
D. Volume occupies space having length, width, and height.
|
A.1 How can fractions be added or subtracted?
A.2 How can fractions be multiplied?
A.3 How can fractions and whole numbers be divided to solve a problem?
B.1 How can fractions be expressed as decimals?
C.1 How are decimal fractions and the place value system related?
D.1 What is volume? How can it be calculated?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Graph points on the coordinate plane to solve real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Coordinate Plane
|
3 - Graph (points on a coordinate plane)
3 - Solve (coordinate plane real-world and mathematical problems)
|
3 - I can graph points on a coordinate plane.
3 - I can solve real-world and mathematical problems involving coordinate planes.
|
A. Coordinate planes can be used to solve real-world and mathematical problems.
|
A.1 How can points be graphed on a coordinate plane?
A.2 How can real-world and mathematical problems be solved using coordinate planes?
|
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Coordinate System
|
3 - Use (pair of perpendicular number lines called axes)
2 - Define (coordinate system, axis, origin, coordinates)
2 - Understand (relationship between ordered pairs, x-axis, x-coordinate, y-axis, y-coordinate)
|
3 - I can use a pair of perpendicular number lines, or axes, to form a coordinate plane.
2 - I can define a coordinate system using the definitions of axis, origin, and coordinates.
2 - I can understand the relationship between ordered pairs and the x-axis, x-coordinate, y-axis, and y-coordinate.
|
A. To use a coordinate plane, one must understand how to use ordered pairs.
|
A.1 What are the features of a coordinate plane? How do you make a coordinate plane?
A.2 How can points be plotted on a coordinate plane?
|
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Coordinate Plane
Problems
|
2 - Represent (real-world and mathematical problems, coordinate plane)
3 - Graph (points, first quadrant, coordinate plane)
3 - Interpret (coordinate values of points, situational context)
|
2 - I can represent real-world and mathematical problems with the first quadrant of the coordinate plane.
3 - I can graph points in the first quadrant of the coordinate plane.
3 - I can interpret coordinate values of points in the context of the situation.
|
A. By graphing points and interpreting their values on a coordinate plane, real-world and mathematical problems can be solved.
|
A.1 What is a coordinate plane? How can real-world and mathematical problems be solved using a coordinate plane?
|
Classify two-dimensional figures into categories based on their properties.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional Figures
|
3 - Classify (two-dimensional figure categories by their properties)
|
3 - I can classify two-dimensional figures into categories based on their properties.
|
A. Two-dimensional figures can be classified based on their properties.
|
A.1 How can two-dimensional figures be classified based on their properties?
|
Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional Figures
|
2 - Understand (two- dimensional figure category attributes are the same for its subcategories)
|
2 - I can understand that attributes belonging to a two-dimensional figure category also belong to all subcategories of that category.
|
A. Attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
|
A.1 How do attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category?
|
Classify two-dimensional figures in a hierarchy based on properties.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Two-Dimensional Figures
|
3 - Classify (two-dimensional figures in a hierarchy)
|
3 - I can classify two-dimensional figures in a hierarchy based on properties.
|
A. Two-dimensional figures can be classified in a hierarchy based on their properties.
|
A.1 What is a hierarchy? How can two-dimensional figures be classified in a hierarchy?
|
In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Fractions
Mathematical terms
Thinking Strategy
|
2 - Connect (ratio and rate to whole number multiplication and division)
3 - Use (ratio and rate concepts to solve problems)
2 - Understand (division of fractions)
2 - Understand (rational number system, negative numbers)
3 - Write (expressions and equations)
2 - Interpret (expressions and equations)
3 - Use (expressions and equations)
2 - Understand (statistical thinking)
|
2 - I can connect ratio and rate to whole number multiplication and division.
3 - I can use ratio and rate concepts to solve problems.
2 - I can understand division of fractions.
2 - I can understand rational numbers including negative numbers.
3 - I can write expressions and equations.
3 - I can interpret expressions and equations.
3 - I can use expressions and equations.
2 - I can think statistically.
|
A. Rate is a ratio.
B. Fractions can be divided.
C. Rational numbers can be used in fractions.
D. Mathematical terms can be written as expressions or equations.
E. Thinking statistically helps to solve problems.
|
A.1 What is rate? How is rate and ratio related?
A.2 How is multiplication and division used with ratio and rate problems?
B.1 How can fractions be divided?
C.1 What are rational numbers? How can one determine if a number is rational?
C.2 What are negative numbers? How can negative numbers be rational?
D.1 What is a mathematical expression? What is a mathematical equation? How are they alike? Different?
D.2 How are mathematical expressions and equations written, interpreted, and used?
E.1 What is statistics? How can it be used in thinking to solve problems?
|
Geometry
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Solve real-world and mathematical problems involving area, surface area, and volume.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Problems
|
3 - Solve (real-world and mathematical problems, area, surface area, and volume)
|
3 - I can solve real-world and mathematical problems involving area, surface area, and volume.
|
A. Real-world and mathematical problems involving area, surface area, and volume can be solved.
|
A.1 How can real-world and mathematical problems involving area, surface area, and volume be solved?
|
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Area Problem Techniques
Problems
|
2 - Find (area of right triangles, other triangles, special quadrilaterals, polygons by composing into rectangles)
2 - Find (area of right triangles, other triangles, special quadrilaterals, polygons by decomposing into triangles and other shapes)
3 - Apply (Area problem techniques, real-world and mathematical problems)
|
2 - I can find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing them into rectangles.
2 - I can find the area of right triangles, other triangles, special quadrilaterals, and polygons by decomposing them into triangles and other shapes.
3 - I can apply composing and decomposing techniques to find the area in solving real-world and mathematical problems.
|
A. Shapes can be composed into rectangles or decomposed into triangles to help solve real-world or mathematical problems.
|
A.1 How can area be found by composing or decomposing shapes?
|
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Find (volume of a right rectangular prism by packing with unit cubes)
3 - Show (volume by packing is equal to multiplying edge lengths)
3 - Apply (V = l w h and V = b h to find volumes in the context of real-world and mathematical problems)
|
2 - I can find volume of a right rectangular prism by packing with unit cubes.
3 - I can show how the volume of a right rectangular prism is the same by packing unit cubes or multiplying the edge lengths.
3 - I can apply V = l w h and V = b h to find volumes in the context of real-world and mathematical problems.
|
A. The volume of a rectangular prism can be determined using unit cubes and multiplying the prism's edge lengths.
|
A.1 How can the volume of a right rectangular prism be determined?
A.2 What are the formulas for finding the volume of a right rectangular prism? How can they be used for solving real-world problems?
|
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Coordinate Plane
Problems
|
3 - Draw (polygons in the coordinate plane given vertice coordinates)
3 - Use (coordinates on a coordinate plane)
2 - Find (length of a side joining points with same first coordinate)
2 - Find (length of a side joining points with same second coordinate)
3 - Apply (techniques of finding lengths in a coordinate system to solve real-world and mathematical problems)
|
3 - I can draw polygons in the coordinate plane given vertice coordinates.
3 - I can use coordinates on a coordinate plane.
2 - I can find the length of a side joining points with same first coordinate.
2 - I can find the length of a side joining points with same second coordinate.
3 - I can apply techniques of finding lengths in a coordinate system to solve real-world and mathematical problems.
|
A. Figures can be drawn and measured on a coordinate system.
|
A.1 How are polygons drawn on a coordinate system?
A.2 How can the side of a polygon be measured on a coordinate system?
|
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Figures
Techniques
Problem context
|
2 - Represent (Three dimensional figure)
3 - Use (Nets)
2 - Find (Surface area)
3 - Apply (Techniques)
3 - Solve (Real world problems)
3 - Solve (Mathematical problems)
|
2 - I can represent three dimensional figures using nets made up of rectangles and triangles.
3 - I can use nets made up of rectangles and triangles to find the surface area of a three dimensional figure.
2 - I can find the surface area of a three dimensional figure by using a net.
3 - I can apply the technique of using a net to solve problems.
3 - I can solve real world problems involving finding the surface area of a three dimensional figure by using nets.
3 - I can solve mathematical problems involving finding the surface area of a three dimensional figure by using nets.
|
A. Nets of three dimensional figures can be used to solve real world problems involving surface area.
|
A.1 What is a three dimensional figure?
A.2 What is a net?
A.3 How does one design a net for a three dimensional figure?
A.4 How does one find the surface area of a three dimensional figure by using the net made up of triangles and rectangles?
A.5 How does one use nets to solve real world problems involving finding the surface area of a three dimensional figure?
A.6 How does one use nets to solve mathematical problems involving finding the surface area of a three dimensional figure?
|
In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Mathematical Relationships
Operations
Shapes
Problems
Populations
|
2 - Understand (proportional relationships)
3 - Apply (proportional relationships)
2 - Understand (operations with rational numbers)
3 - Work (with expressions and linear equations)
3 - Solve (problems involving scale drawings and informal geometric constructions)
3 - Work (with two- and three-dimensional shapes)
3 - Solve (problems involving area, surface area, and volume)
3 - Infer (about populations based on samples)
|
2 - I can understand proportional relationships.
3 - I can apply proportional relationships.
2 - I can understand operations with rational numbers.
3 - I can work with expressions and linear equations.
3 - I can solve problems involving scale drawings and informal geometric constructions.
3 - I can work with two- and three-dimensional shapes.
3 - I can solve problems involving area, surface area, and volume.
3 - I can infer about populations based on samples.
|
A. Proportional relationships can be used to solve real-world and mathematical problems.
B. Rational numbers can be used in a variety of mathematical operations.
C. Mathematical terms can be written as expressions or equations.
D. Problems involving two- and three-dimensional shapes can be solved using scale drawings and informal geometric constructions.
E. Inferences can be made about populations using samples.
|
A.1 What are proportional relationships? How can they be used to solve problems?
B.1 What is a rational number? What are the different operations that can be done on rational numbers?
C.1 What is a linear equation? How is it different from other expressions?
D.1 How can problems be solved involving scale drawings and informal geometric constructions?
D.2 How can area, surface area, and volume problems be solved involving two- and three-dimensional shapes?
E.1 How can inferences be made about populations using samples?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Draw, construct, and describe geometrical figures and describe the relationships between them.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Geometric
|
3 - Draw (geometrical figures)
3 - Construct (geometrical figures)
1 - Describe (geometrical figures)
2 - Describe (relationships between geometrical figures)
|
3 - I can draw geometrical figures.
3 - I can construct geometrical figures.
1 - I can describe geometrical figures.
2 - I can describe relationships between geometrical figures.
|
A. Geometrical figures can be drawn, constructed, and described.
B. Relationships between geometric figures can be identified and described.
|
A.1 How can geometrical figures be drawn, constructed, or described?
B.1 How can relationships between geometric figures be identified? How can these relationships be described?
|
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
3 - Solve (problems involving scale drawings of geometric figures)
3 - Compute (actual lengths and areas from a scale drawing)
3 - Reproduce (scale drawing at a different scale)
|
3 - I can solve problems involving scale drawings of geometric figures.
3 - I can compute actual lengths and areas from a scale drawing.
3 - I can reproduce a scale drawing at a different scale.
|
A. Actual lengths and areas can be computed from scale drawings.
B. Scaled drawings can be scaled up or down into larger or smaller drawings.
|
A.1 What information can be found from a scale drawing? How can this information be computed?
B.1 How can a scaled drawing be increased or decreased in size using mathematics?
|
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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Drawings
Geometric shapes
Triangles
Conditions of Constructed Figure
|
3 - Draw (Geometric shapes by freehand)
3 - Draw (Geometric shapes by ruler)
3 - Draw (Geometric shapes by technology)
3 - Construct (Triangles)
1 - Notice (Conditions)
|
3 - I can draw geometric shapes by freehand if given enough information about the sides and angles.
3 - I can draw geometric shapes with a ruler if given enough information about the sides and angles.
3 - I can draw geometric shapes with technology if given enough information about the sides and angles.
3 - I can construct triangles is given three measures of sides or angles.
1 - I can notice when the conditions given determine a unique triangle, more than one triangle or no triangle.
|
A .One can draw a geometric shape when given conditions either by freehand, ruler and protractor, or with technology.
B. One should notice when the conditions given for a triangle determine a unique triangle, more than one triangle or no triangle.
|
A.1 How does one freehand geometric shapes when given conditions for the measures of the angles and sides?
A.2 How does one use a ruler and a protractor to draw a geometric shape when given conditions for the measures of the angles and sides?
A.3 How does one use technology to draw a geometric shape when given conditions for the measures of the angles and sides?
B.1 What conditions for the angles and side lengths of a triangle indicate a unique triangle?
B.2 What conditions for the angles and side lengths of a triangle indicate more than one triangle?
B.3 What conditions for the angles and side lengths of a triangle indicate that no triangle could be drawn.
|
Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Describe (Two dimensional figure)
2 - Slice (Two dimensional figure)
|
2 - I can describe the two dimensional figure that results from slicing a three dimensional figure.
2 - I can mentally picture the results of slicing a three dimensional figure.
|
A. One can mentally picture the two dimensional figure that results from slicing a three dimensional figure and then can describe those results.
|
A.1 What is a two dimensional figure?
A.2 What are the names of the two dimensional figures?
A.3 What is a three dimensional figure?
A.4 What are the names of the three dimensional figures?
A.5 How does one mentally slice a three dimensional figure?
A.6 How does one describe a three dimensional figure?
|
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Problems
Figure Features
|
3 - Solve (real-life and mathematical problems involving angle measure, area, surface area, and volume)
|
3 - I can solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
|
A. Real-world and mathematical problems involving angle measure, area, surface area, and volume can be solved.
|
A.1 How can real-world and mathematical problems involving angle measure, area, surface area, and volume be solved?
|
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Problems
Relationship between:
|
1 - Know (Formula for area of a circle)
1 - Know (Formula for circumference of a circle)
3 - Use (Formula of a circle)
3 - Use (Formula of the area of a circle)
3 - Solve (Problems)
2 - Give (Derivation of relationship between area and circumference of a circle)
|
1 - I can state the formula for the area of a circle.
1 - I can state the formula for the circumference of a circle.
3 - I can use the formula for circumference of a circle.
3 - I can use the formula for the area of a circle.
3 - I can solve problems about area and circumference of circles.
3 - I can give the derivation of the relationship between area and circumference of a circle.
|
A. One can memorize and then use the formulas for the area and circumference of a circle to solve problems.
B. One can also give an informal derivation of the relationship between the circumference and area of a circle.
|
A.1 What is the formula for the area of a circle?
A.2 What is the formula for the circumference of a circle?
A.3 How does one use the formula for area to solve a problem?
A.4 How does one use the formula for circumference to solve a problem?
B.1 How does one derive the relationship between the circumference and area of a circle?
|
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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Problems
|
3 - Use (Facts about supplementary angles)
3 - Use (Facts about complementary angles)
3 - Use (Facts about vertical angles)
3 - Use (Facts about adjacent angles)
3 - Write (Equation)
3 - Solve (Equation)
|
3 - I can use facts about supplementary angles.
3 - I can use facts about complementary angles.
3 - I can use facts about vertical angles.
3 - I can use facts about adjacent angles.
3 - I can write a simple equation for an unknown angle in a figure.
3 - I can solve a simple equation for an unknown angle in a figure.
|
A. One can write and solve an equation for an unknown angle in a figure using facts about supplementary, complementary, vertical and adjacent angles.
|
A.1 What is a supplementary angle?
A.2 What is a complementary angle?
A.3 What is a vertical angle?
A.4 What is an adjacent angle?
A.5 What facts does one know about supplementary angles?
A.6 What facts does one know about complementary angles?
A.7 What facts does one know about vertical angles?
A.8 What facts does one know about adjacent angles?
A.9 How does one write a simple equation for an unknown angle in a figure?
A.10 How does one solve a simple equation for an unknown angle in a figure?
|
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Problems
Measures
Figures
|
3 - Solve (Problems involving area of a triangle)
3 - Solve (Problems involving area of a quadrilateral)
3 - Solve (Problems involving area of a polygon)
3 - Solve (Problems involving volume of a cube)
3 - Solve (Problems involving volume of a right prism)
3 - Solve (Problems involving surface area of a cube)
3 - Solve (Problems involving surface area of a right prism)
|
3 - I can solve real life and mathematical problems involving the area of a triangle.
3 - I can solve real life and mathematical problems involving the area of a quadrilateral.
3 - I can solve real life and mathematical problems involving the area of a polygon.
3 - I can solve real life and mathematical problems involving the volume of a cube.
3 - I can solve real life and mathematical problems involving the volume of a right prism.
3 - I can solve real life and mathematical problems involving the surface area of a cube.
3 - I can solve real life and mathematical problems involving the surface area of a right prism.
|
A. One can solve real life or mathematical problems involving the area of a triangle, quadrilateral or polygon, the volume of a cube or a right prism, or the surface area of a cube or right prism.
|
A.1 What is a two dimensional figure?
A.2 What is a three dimensional figure?
A.3 What are the different polygons?
A.4 What are right prisms?
A.5 How does one find the area a triangle?
A.6 How does one find the area of a quadrilateral?
A.7 How does one find the area of a polygon?
A.8 How does one find the volume of a right prism?
A.9 How does one find the surface area of a right prism?
A.10 How does one solve problems involving area?
A.11 how does one solve problems involving volume?
A.12 How does one solve problems involving surface area?
|
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Expressions
Equations
Functions
|
2 - Reason (about expressions)
2 - Reason (about equations)
3 - Model (association in bivariate data with linear equation)
2 - Solve (linear equations)
2 - Solve (systems of linear equations)
2 - Understand (functions)
3 - Describe (quantitative relationships with functions)
4 - Analyze (two-dimensional space)
4 - Analyze (three-dimensional space)
4 - Analyze (two-dimensional figures)
4 - Analyze (three-dimensional figures)
3 - Use (distance, angles, similarity, and congruence)
2 - Understand (Pythagorean Theorem)
3 - Apply (Pythagorean Theorem)
|
2 - I can reason about expressions.
2 - I can reason about equations.
3 - I can model association in bivariate data with linear equation.
2 - I can solve linear equations.
2 - I can solve systems of linear equations.
2 - I can understand functions.
3 - I can describe quantitative relationships with functions.
4 - I can analyze two-dimensional space.
4 - I can analyze three-dimensional space.
4 - I can analyze two-dimensional figures.
4 - I can analyze three-dimensional figures.
3 - I can use distance, angles, similarity, and congruence.
2 - I can understand the Pythagorean Theorem.
3 - I can apply the Pythagorean Theorem.
|
A. Equations can be used alone or in systems to solve real world problems.
B. Functions can be used to describe numerical relationships and can be modeled using a graph.
C. Two-and three-dimensional figures can be congruent or similar.
|
A.1 What is an equation?
A.2 How are equations used?
B.1 What is a function?
B.2 How can I represent functions?
C.1 what does it mean for figures to be congruent?
C.2 What does it mean for figures to be similar?
|
Geometry (G)
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Domain
|
None Available | None Available | None Available | None Available |
Understand congruence and similarity using physical models, transparencies, or geometry software.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Physical models
Transparencies
Geometry software
|
1 - Understand (Congruence)
1 - Understand (Similarity)
|
1 - I understand congruence using physical models, transparencies or geometry software.
1 - I understand similarity using physical models, transparencies or geometry software.
|
A. Congruence and similarity can be more clearly understood by using physical models, transparencies and geometry software.
|
A.1 What is the definition of congruent?
A.2 What is the definition of similarity?
A.3 How does one determine congruence using physical models?
A.4 How does one determine similarity using physical models?
A.5 How does one determine congruence using transparencies?
A.6 How does one determine similarity using transparencies?
A.7 How does one determine congruence using geometry software?
A.8 How does one determine similarity using geometry software?
|
Verify experimentally the properties of rotations, reflections, and translations:
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Properties
Angles [b]
Lines [c]
|
5 - Verify (Properties of rotations)
5 - Verify (Properties of reflections)
5 - Verify (Properties of translations)
2 - Understand (lines taken to lines of same length [a])
2 - Understand (line segments taken to line segments of same length [a])
2 - Understand (angles taken to angles of same measure [b])
2 - Understand (parallel lines taken to parallel lines [c])
|
5 - I can verify experimentally the properties of rotations.
5 - I can verify experimentally the properties of reflections.
5 - I can verify experimentally the properties of translations.
2 - I can understand that lines are taken to lines of same length [a].
2 - I can understand that line segments are taken to line segments of same length [a].
2 - I can understand that angles are taken to angles of same measure [b].
2 - I can understand that parallel are lines taken to parallel lines [c].
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A. The properties of rotations, reflections and translations can be verified experimentally.
B. Rotations, reflections, and translations all preserve figure congruence.
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A.1 What is a rotation?
A.2 What are the properties of rotations?
A.3 What is a reflection?
A.4 What are the properties of reflections?
A.5 What is a translation?
A.6 What are the properties of translations?
A.7 How can one experimentally verify the properties of rotations?
A.8 How can one experimentally verify the properties of reflections?
A.9 How can one experimentally verify the properties of translations?
B.1 What effect does rotation, reflection, or translation have on the figure being transformed?
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Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Figures
Sequence
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1 - Understand (Congruence between two figures)
2 - Describe (Sequence of rotations, reflections and translations)
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1 - I understand that a two dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections or translations.
2 - I can describe the sequence of rotations, reflections and translations that exhibits congruence between two figures.
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A. Two different two dimensional figures are congruent only if the second figure can be obtained from the first figure through a sequence of rotations, reflections and translations.
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A.1 What makes two figures congruent?
A.2 What is a rotation?
A.3 What is a reflection?
A.4 What is a translations?
A.5 What is a sequence?
A.6 What is a two dimensional figure?
A.7 How does one find the sequence of rotations, reflections or translations that exhibits congruence between two figures?
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Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Effects
|
2 - Describe (Effect of dilations)
2 - Describe (Effect of translations)
2 - Describe (Effect of reflections)
2 - Describe (Effect of rotations)
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2 - I can describe the effect of dilations on two dimensional figures using coordinates.
2 - I can describe the effect of translations on two dimensional figures using coordinates.
2 - I can describe the effect of reflections on two dimensional figures using coordinates.
2 - I can describe the effect of rotations on two dimensional figures using coordinates.
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A. The effect of dilations, translations, rotations and reflections on a two dimensional figure can be described by using coordinates.
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A.1 What does the term effect mean?
A.2 What is a dilation?
A.3 What is a translation?
A.4 What is a rotation?
A.5 What is a reflection?
A.6 What is a two dimensional figure?
A.7 What is a coordinate?
A.8 How does one describe the effect of dilations, translations, rotations, and reflections of two dimensional figures using coordinates?
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Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Figures
Sequence
|
1 - Understand (Similarity between two figures)
2 - Describe (Sequence of rotations, reflections, translations and dilations)
|
1 - I understand that a two dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, dilations and translations.
2 - I can describe the sequence of rotations, reflections, translations and dilations that exhibits similarity between two figures.
|
A. A two dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, dilations and translations.
|
A.1 What makes two figures similar?
A.2 What is a rotation?
A.3 What is a reflection?
A.4 What is a translation?
A.5 What is a dilation?
A.6 What is a sequence?
A.7 What is a two dimensional figure?
A.8 How does one find the sequence of rotations, reflections or translations that exhibits congruence between two figures?
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Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Arguments
Facts of triangles
Facts of parallel lines/transversal
Similarity
|
3 - Use (Informal arguments)
3 - Establish (Facts about angle sum in triangles)
3 - Establish (Facts about exterior angles of triangles)
3 - Establish (Facts about angles created when parallel lines are cut by a transversal)
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3 - I can use informal arguments to establish facts.
3 - I can establish facts about the angle sum in triangles.
3 - I can establish facts about the exterior angles of triangles.
3 - I can establish facts about the angles created when parallel lines are cut by a transversal.
3 - I can establish facts about the angle/angle criteria for similarity of a triangle.
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A. Informal arguments can be used to establish facts about the angle sum and exterior angles of triangles, about the angles created when parallel lines are cut by a transversal, and about the angle/angle criterion for similarity of triangles.
|
A.1 What is an informal argument?
A.2 What facts can be established about the sum of the angles in a triangle?
A.3 What facts can be established about the exterior angles of triangles?
A.4 What are parallel lines?
A.5 What is a transversal?
A.6 What facts can be established about the angles formed when parallel lines are cut by a transversal?
A.7 What facts can be established about the angle/angle criteria for triangle similarity?
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Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove theorems about lines and angles.
Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Prove theorems about triangles.
Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Prove theorems about parallelograms.
Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Understand similarity in terms of similarity transformations
Verify experimentally the properties of dilations given by a center and a scale factor:
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems about triangles.
Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
(+) Construct a tangent line from a point outside a given circle to the circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Translate between the geometric description and the equation for a conic section
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Derive the equation of a parabola given a focus and directrix.
(+) Derive the equations of ellipses and hyperbolas given foci and directrices.
Use coordinates to prove simple geometric theorems algebraically
Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Explain volume formulas and use them to solve problems
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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