10th Grade Math

 

Measurement Geometry and Spatial Sense Patterns, Functions and Algebra Data Analysis and Probability


 

Number, Number Sense and Operations

(Based on State of Ohio
Curriculum Standards)

1. Connect physical, verbal and symbolic representations of irrational numbers; e.g., construct square root of 2 as a hypotenuse or on a number line.

 
  1. Comparing and Ordering Integers-Compare and order integers using draggable points on a number line. Also explore the opposites of numbers on the number line.
  2. Square Roots-Explore the meaning of square roots using an area model.  Use the side length of a square to find the square root of a decimal number or a whole number.
     

2. Explain the meaning of the nth root.

  1. Math Dictionary animation for kids  This is an outstanding website allowing you to explore a comprehensive view of math with many examples  4 Star

3. Use factorial notation and computations to represent and solve problem situations involving arrangements.

 

4. Approximate the nth root of a given number greater than zero between consecutive integers when n is an integer; e.g., the 4th root of 50 is between 2 and 3.

 

Measurement

1. Explain how a small error in measurement may lead to a large error in calculated results.

 

2. Calculate relative error.

 

3. Explain the difference between absolute error and relative error in measurement.

 

4. Give examples of how the same absolute error can be problematic in one situation but not in another; e.g., compare “accurate to the nearest foot” when measuring the height of a person versus when measuring the height of a mountain.

 

5. Determine the measures of central and inscribed angles and their associated major and minor arcs.

  1. Chords and Arcs-Explore the relationship between a central angle and the arcs it intercepts. Also explore the relationship between chords and their distance from the circle's center.
     
  2. Inscribing Angles-Resize angles inscribed in a circle.  Investigate the relationship between inscribed angles and the arcs they intercept.

Geometry and Spatial Sense

1. Formally define and explain key aspects of geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median, altitude, midsegment);
c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter);
d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle).

  1. Mathway This a step by step math solver for General Math, Geometry, Algebra, Pre Algebra, Trigonometry, Pre-calculus, Calculus.  A great program to show students step by step directions and also allow a teacher to create a presentation to be used on their LSD projector.  This is a great tool for facilitating and allowing at teacher to better present their instruction.   4 Star 
  2. Classifying Quadrilaterals - Activity A-Apply constraints to a quadrilateral, and then reshape and resize it. Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals.
  3. Classifying Quadrilaterals - Activity B-Apply constraints to a quadrilateral, and then reshape and resize it. Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals.
  4. Triangle Angle Sum - Activity A-Measure the angles of a triangle and find the sum.
  5. Concurrent Lines, Medians, and Altitudes-Explore the relationships between perpendicular bisectors, the circumscribed circle, angle bisectors, the inscribed circle, altitudes, and medians using a triangle that can be resized and reshaped.
  6. Chords and Arcs-Explore the relationship between a central angle and the arcs it intercepts.  Also explore the relationship between chords and their distance from the circle's center.
  7. Circle: Circumference and Area-Resize a circle and compare its radius, circumference, and area.
  8. Inscribing Angles-Resize angles inscribed in a circle.  Investigate the relationship between inscribed angles and the arcs they intercept.
  9. Holiday Snowflake Designer-Fold paper and cut in a certain way to make symmetrical snowflakes with six sides (similar to what can be found in nature) or with eight sides (an easier folding method). This simulation allows you to cut virtual paper on the computer screen with round dot or square dot "scissors" of various sizes before using physical paper.
  10. Reflections-Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated.
  11. Rotations, Reflections and Translations-Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure.
     
 

 

2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane.

 

3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and quadrilaterals;
d. test a conjecture using basic constructions made with a compass and straightedge or technology.

  1. Pythagorean Theorem - Virtual Manipulative
    This online activity allows you to demonstrate the proof of the Pythagorean Theorem using area models that can be manipulated.
  2. Biconditional Statement-Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form.
  3. Conditional Statement-
    Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form.
  4. Geoboard: The Pythagorean Theorem-Build right triangles in an interactive geoboard and build squares on the sides of the triangles to discover the Pythagorean Theorem.
  5. Proving Triangles Congruent-Apply constraints to two triangles. Then drag the vertices of the triangles around and determine which constraints guarantee congruence.
  6. Pythagorean Theorem - Activity B-Explore the Pythagorean Theorem using a dynamic right triangle.  Examine a visual, geometric application of the Pythagorean Theorem, using the areas of squares on the sides of the triangle.
  7. Biconditional Statement-Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form.
  8. Area of Parallelograms - Activity A-Examine and manipulate the graph of a parallelogram or triangle and find its area.
  9. Construct Parallel and Perpendicular Lines-Construct parallel and perpendicular lines using a straightedge and compass. Use step‑by‑step explanations and feedback to develop understanding of the construction.

4. Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software.

  1. Mathway This a step by step math solver for General Math, Geometry, Algebra, Pre Algebra, Trigonometry, Pre-calculus, Calculus.  A great program to show students step by step directions and also allow a teacher to create a presentation to be used on their LSD projector.  This is a great tool for facilitating and allowing at teacher to better present their instruction.   4 Star
  2. Classifying Quadrilaterals - Activity B-Apply constraints to a quadrilateral, and then reshape and resize it.  Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals.
  3. Constructing Congruent Segments and Angles-Construct congruent segments and angles using a straightedge and compass.  Use step‑by‑step explanations and feedback to develop understanding of the construction.
  4. Parallelogram Conditions-Apply constraints to a dynamic quadrilateral.  Then drag its vertices around. Determine which constraints guarantee that the quadrilateral is always a parallelogram.
  5. Special Quadrilaterals-
    Apply constraints to a quadrilateral and determine what type of quadrilateral results.

5. Construct congruent figures and similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software.

  1. Congruent Triangles - Virtual Manipulative
    This online activity allows you to manipulate virtual angles and segments to learn what does and does not guarantee congruent triangles. The methods covered include SSS, SAS, ASA, and SSA
  2. Congruence in Right Triangles-Apply constraints to two right triangles.  Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent.
  3. Proving Triangles Congruent-Apply constraints to two triangles.  Then drag the vertices of the triangles around and determine which constraints guarantee congruence.

6. Identify the reflection and rotation symmetries of two- and three-dimensional figures.

 

7. Perform reflections and rotations using compass and straightedge constructions and dynamic geometry software.

  1. Math Manipulatives - Flips, Slides, and Turns
    This online activity from Encyclopedia Britanica allows the user to place geometric figures and then apply reflections, rotations, and translations to them.
  2. Rotation Transformation - Virtual Manipulative
    This online activity allows you to place pattern blocks on a grid and apply a rotation to them. You can do this on your own, or go through three pre-made activities.
  3. Reflection Transformation - Virtual Manipulative
    This online activity allows you to place pattern blocks on a grid and apply a reflection to them. You can do this on your own, or go through three pre-made activities

8. Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane.

  1. Translation Transformation - Virtual Manipulative
    This online activity allows you to place pattern blocks on a grid and apply a translation to them. You can do this on your own, or go through three pre-made activities.
  2. Rotation Transformation - Virtual Manipulative
    This online activity allows you to place pattern blocks on a grid and apply a rotation to them. You can do this on your own, or go through three pre-made activities.
  3. Reflection Transformation - Virtual Manipulative
    This online activity allows you to place pattern blocks on a grid and apply a reflection to them. You can do this on your own, or go through three pre-made activities.
  4. Rotations, Reflections and Translations-Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure.
  5. Translations-Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation.

9. Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable.

  1. Multiple Transformations - Virtual Manipulative
    This online activity allows you to apply two consecutive transformations to pattern blocks, including translations, reflections, and rotations. You can work on your own, or go through two pre-made activities.
  2. Reflections-Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated.
  3. Rotations, Reflections and Translations-
    Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure.
  4. Translations-Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation.

10. Solve problems involving chords, radii and arcs within the same circle.

 

Patterns, Functions and Algebra

1. Define function formally and with f(x) notation.

  1. Smart Board Algebra Lesson  This lesson has examples and videos.  Smart board manipulativeSB
  2. Algebra Tiles icon   Algebra Tiles Visualize multiplying and factoring algebraic expressions using tiles.   Algebra Tile manipulative
  3. Algebraic Skills: Signed Numbers -  This website will assist you in teaching this concept.
  4. Algebra Balance Scales icon Algebra Balance Scales Solve simple linear equations using a balance beam representation.
  5. Algebra Balance Scales - Negatives icon Algebra Balance Scales - NegativesSolve simple linear equations using a balance beam representation.
  6. Student Resources for Algebra Outstanding web site for teachers and students.
     
 

2. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible number of roots, domain and range.

  1. Cubic Functions: Graphing
    This study guide from Encyclopedia Britannica covers an overview of cubic functions. Includes sections on creating graphs by transformation, using a sign diagram, finding turning points, the factor theorem, graphing a cubic function by factoring, and using a TI-83 graphical calculator.
  2. Cosine Function-Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle.
  3. Cubic Function Activity-Explore a cubic function, along with the associated Center of Rotation.
  4. Functions Involving Square Roots-Compare the graph of a function involving square roots to its equation. Vary the coefficients of the equation and investigate how the graph changes in response.
  5. Inequalities Involving Absolute Values-Solve an inequality involving absolute values using a graph of the absolute-value function.
  6. Radical Functions-Compare the graph of a radical function to its equation. Vary the terms of the equation.
  7. Sine Function-Compare the graph of the sine function with the graph of the angle on the unit circle.  Drag a point along the sine curve and see the corresponding angle on the unit circle.
  8. Tangent Function-Compare the graph of the tangent function with the graph of the angle on the unit circle.  Drag a point along the tangent curve and see the corresponding angle on the unit circle.

3. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height.

  1. Mathway This a step by step math solver for General Math, Geometry, Algebra, Pre Algebra, Trigonometry, Pre-calculus, Calculus.  A great program to show students step by step directions and also allow a teacher to create a presentation to be used on their LSD projector.  This is a great tool for facilitating and allowing at teacher to better present their instruction.   4 Star
  2. Perimeter, Circumference, and Area - Activity B-Resize a rectangle, circle, or square and find its perimeter or circumference and area.

4. Use algebraic representations and functions to describe and generalize geometric properties and relationships.

  1. Ellipse - Activity A-Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response.
  2. Hyperbola - Activity A-Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola.
  3. Mathway This a step by step math solver for General Math, Geometry, Algebra, Pre Algebra, Trigonometry, Pre-calculus, Calculus.  A great program to show students step by step directions and also allow a teacher to create a presentation to be used on their LSD projector.  This is a great tool for facilitating and allowing at teacher to better present their instruction.   4 Star
     

5. Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and solutions.

  1. Modeling One-Step Equations - Activity A-Solve one‑step equations using an area model.  Drag tiles into two bins, each representing a side of the equation, and isolate an x‑tile on one side.
  2. Modeling and Solving Two-Step Equations-Solve a two‑step equation using a cup‑and‑counter model. Use step‑by‑step feedback to diagnose and correct incorrect steps.
  3. Solving Equations By Graphing Each Side-Solve an equation by graphing each side and finding the intersection of the lines. Vary the coefficients in the equation and explore how the graph changes in response.
  4. Solving Linear Inequalities using Addition and Subtraction-
    Solve a linear inequality. Graph the solution on a dynamic number line.
  5. Solving Linear Inequalities using Multiplication and Division-
    Solve an inequality involving multiplication and division. Graph the solution on a number line.
  6. Solving Two-Step Equations-Choose the correct steps to solve a two-step equation. Use the feedback to diagnose incorrect steps.
  7. Square Roots-Explore the meaning of square roots using an area model. Use the side length of a square to find the square root of a decimal number or a whole number.

6. Solve equations and inequalities having rational expressions as coefficients and solutions.

  1. Modeling One-Step Equations - Activity A-Solve one‑step equations using an area model. Drag tiles into two bins, each representing a side of the equation, and isolate an x‑tile on one side.
  2. Modeling and Solving Two-Step Equations-Solve a two‑step equation using a cup‑and‑counter model.  Use step‑by‑step feedback to diagnose and correct incorrect steps.
  3. Solving Linear Inequalities using Addition and Subtraction-
    Solve a linear inequality. Graph the solution on a dynamic number line.
  4. Solving Linear Inequalities using Multiplication and Division-Solve an inequality involving multiplication and division. Graph the solution on a number line.
  5. Solving Two-Step Equations-Choose the correct steps to solve a two-step equation. Use the feedback to diagnose incorrect steps.

7. Solve systems of linear inequalities.

  1. Linear Programming - Activity A-Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response.
  2. Solving Linear Inequalities using Addition and Subtraction-
    Solve a linear inequality. Graph the solution on a dynamic number line.
  3. Solving Linear Inequalities using Multiplication and Division-Solve an inequality involving multiplication and division. Graph the solution on a number line.
  4. Systems of Linear Inequalities (Slope-intercept form) - Activity A-Compare a system of linear inequalities to its graph.  Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response.

8. Graph the quadratic relationship that defines circles.

  1. Circles-Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response.

9. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals.

  1. Slope - Activity B-Explore the slope of a line, and learn how to calculate slope. Adjust the line by moving points that are on the line, and see how its slope changes.

10. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.

  1. Exponential Functions - Activity A-Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function.
  2. Functions Involving Square Roots-Compare the graph of a function involving square roots to its equation.
  3. Linear Functions-Determine if a relation is a function from the mapping diagram, ordered pairs, or graph.
  4. Quadratic and Absolute Value Functions-Compare the graph of a quadratic or absolute‑value function to its equations. Vary the coefficients of the equation and explore how the graph changes in response.
  5. Quadratics in Factored Form-Investigate the factors of a quadratic through its graph and through its equation.  Vary the roots of the quadratic and examine how the graph and the equation change in response.
  6. Quadratics in Polynomial Form - Activity A-Compare the graph of a quadratic to its equation in polynomial form. Vary the coefficients of the equation and explore how the graph changes in response.
  7. Radical Functions-Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation.
  8. Roots of a Quadratic-Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane.

11. Solve real-world problems that can be modeled, using systems of linear equations and inequalities.

  1. Linear Programming - Activity A-Use the graph of the feasible region to find the maximum or minimum value of the objective function.
  2. Modeling Linear Systems - Activity A-Experiment with a system of two lines representing a cat‑and‑mouse chase.  Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.
  3. Special Types of Solutions to Linear Systems-Compare a system of equations in standard form to its graph. Examine the graph and table of values. Determine the solutions to the system.
  4. Systems of Linear Inequalities (Slope-intercept form) - Activity A-Compare a system of linear inequalities to its graph. Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response.

12. Describe the relationship between slope of a line through the origin and the tangent function of the angle created by the line and the positive x-axis.

  1. Slope - Activity B-Explore the slope of a line, and learn how to calculate slope. Adjust the line by moving points that are on the line, and see how its slope changes.
  2. Tangent Function-Compare the graph of the tangent function with the graph of the angle on the unit circle.
  3. Tangent Ratio-Reshape and resize a right triangle and examine how the tangent of angle A changes.

Data Analysis and Probability

1. Describe measures of center and the range verbally, graphically and algebraically.

  1. Mathway This a step by step math solver for General Math, Geometry, Algebra, Pre Algebra, Trigonometry, Pre-calculus, Calculus.  A great program to show students step by step directions and also allow a teacher to create a presentation to be used on their LSD projector.  This is a great tool for facilitating and allowing at teacher to better present their instruction.   4 Star
  2. Box-and-Whisker Plots-Construct a box‑and‑whisker plot to match a line plots, and construct a line plot to match a box‑and‑whisker plots.
  3. Describing Data Using Statistics-Investigate the mean, median, mode, and range of a data set through its graph.
  4. Line Plots-Build a data set using a line plot. Find its mean, median, mode, and range. Change the data values by dragging the points, and watch how the mean, median, mode, and range change (or, in some cases, do not change).
  5. Mean, Median and Mode-Build a data set and find the mean, median, and mode. Explore the mean, median, and mode illustrated as frogs on a seesaw, frogs on a scale, and as frogs stacked under a bar of variable height.

2. Represent and analyze bivariate data using appropriate graphical displays (scatterplots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology.

  1. Scatterplot Activity
    This interactive activity from Encyclopedia Britannica is a statistical tool to explore the relationships between two data sets. Ranges can be set independently and data points entered by hand or loaded from sample sets. Displays an updated regression line and summary statistics.
  2. Scatterplot - Virtual Manipulative
    This online activity allows the user to create scatterplot by adding points to a grid, and then calculates and displays the line of best fit. The grid size can be changed fit any activity, and one suggested activity is explained on the site.
  3. Scatterplots - Excel activity
    This is an Excel spreadsheet activity for individual students or the whole class. As the user enters ordered pairs, the spreadsheet graphs a scatterplot, creates a line of best fit, and displays its equation (in linear or polynomial form).
  4. Box-and-Whisker Plots-Construct a box‑and‑whisker plot to match a line plots, and construct a line plot to match a box‑and‑whisker plots.
  5. Correlation-Explore the relationship between the correlation coefficient of a data set and its graph.
  6. Histograms-Change the values in a data set and examine how the dynamic histogram changes in response.
  7. Populations and Samples-Compare sample distributions drawn from population distributions. Predict characteristics of a population distribution based on a sample distribution and examine how well a small sample represents a given population.
  8. Scatter Plots - Activity A-Examine the scatter plot for a random data set with negative or positive correlation.
  9. Solving Using Trend Lines-Examine the scatter plots for a data related to weather at different latitudes. The Gizmo includes three different data sets, one with negative correlation, one positive, and one with no correlation. Compare the least squares best‑fit line.

3. Display bivariate data where at least one variable is categorical.

  1. Box-and-Whisker Plots-Construct a box‑and‑whisker plot to match a line plots, and construct a line plot to match a box‑and‑whisker plots. Manipulate the line plot and examine how the box‑and‑whisker plot changes. Then manipulate the box‑and‑whisker plot and examine how the line plot changes.
  2. Describing Data Using Statistics-Investigate the mean, median, mode, and range of a data set through its graph.
  3. Histograms-Change the values in a data set and examine how the dynamic histogram changes in response.
  4. Line Plots-Build a data set using a line plot. Find its mean, median, mode, and range. Change the data values by dragging the points, and watch how the mean, median, mode, and range change (or, in some cases, do not change).
  5. Scatter Plots - Activity A-Examine the scatter plot for a random data set with negative or positive correlation.
  6. Stem-and-Leaf Plots-Build a data set and compare the line plot of the data set to the stem‑and‑leaf plot.

4. Identify outliers on a data display; e.g., use interquartile range to identify outliers on a box-and-whisker plot.

  1. Statistics - Deviation from the Mean
    This study guide from Encyclopedia Britannica covers several lessons on stats such as range, interquartile range, box and whisker plots, and more.
  2. Box Plot and Histogram - Virtual Manipulative
    This online interactive activity generates a box plot and a histogram for data provided. You can choose from several pre-made sets of topical data, or you can enter new data or change the data there. Several measures of the data are given as well.
  3. Box-and-Whisker Plots-Construct a box‑and‑whisker plot to match a line plots, and construct a line plot to match a box‑and‑whisker plots. Manipulate the line plot and examine how the box‑and‑whisker plot changes. Then manipulate the box‑and‑whisker plot and examine how the line plot changes.

5. Provide examples and explain how a statistic may or may not be an attribute of the entire population; e.g., intentional or unintentional bias may be present.

 

6. Interpret the relationship between two variables using multiple graphical displays and statistical measures; e.g., scatterplots, parallel box-and-whisker plots, and measures of center and spread.

  1. Scatterplots - Excel activity
    This is an Excel spreadsheet activity for individual students or the whole class. As the user enters ordered pairs, the spreadsheet graphs a scatterplot, creates a line of best fit, and displays its equation (in linear or polynomial form).
  2. Mean, Median and Mode-Build a data set and find the mean, median, and mode. Explore the mean, median, and mode illustrated as frogs on a seesaw, frogs on a scale, and as frogs stacked under a bar of variable height.

7. Model problems dealing with uncertainty with area models (geometric probability).

  1. Geometric Probability - Activity A-Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it

8. Differentiate and explain the relationship between the probability of an event and the odds of an event, and compute one given the other.

  1. Binomial Probabilities-
    Find the probability of a number of successes or failures in a binomial experiment using a tree diagram, a bar graph, and direct calculation.
  2. Geometric Probability - Activity A-Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it