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Older Resources. The resources on this page have been aligned with the 2005–06 revised K–12 mathematics TEKS. However, they have not been fully updated with new material.

For fully updated versions of these activities, please consider purchasing Mathematics Standards in the Classroom.

(8.1) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations.

(8.1.a) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations. The student is expected to compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals.

Clarifying Activity with Assessment Connections

The teacher posts an unlabeled, classroom-size number line marked with the benchmarks -2, -1, -1/2, 0, 1/2, 1, and 2. The teacher provides students with a variety of rational numbers (in fractional, decimal, and percentage form) written on notecards. Students put the notecards in the appropriate places on the number line.

Assessment Connections

Questioning . . .

Open with . . .

• What strategies could you use to help you locate these numbers on the number line?

Probe further with . . .

• Which of these numbers are greater than zero? Less than zero?
• Which numbers are between 0 and 1? How do you know?
• Which are between 0 and -1? How do you know?
• Which are between 1 and 2? How do you know?
• Tell me a number that falls between these two numbers.
• Tell me a fraction between these two numbers.
• Tell me a decimal between these two numbers.
• Tell me a percentage between these two numbers.
• Is there a way to locate the equivalent of 100% on the number line?
• Is there a way to locate the equivalent of 40% on the number line?

Listen for . . .

• Can the student describe numbers using multiple representations to help locate their positions on the line? (For example, .25 = 1/4.)
• Does the student use benchmarks to help determine locations for numbers? (For example, 3/8 is less than 1/2; 40% is less than 50%, etc.)
• Does the student convert to other representations—such as changing fractions to decimals—to determine placement?

Look for . . .

• Does the student place benchmarks at the appropriate points?
• Does the student place positive and negative numbers on the appropriate side of zero on the number line?
• Does the student recognize that the negative value of a number is the same distance away from zero as its positive value?
• Does the student recognize that every value other than zero has an opposite value that could be assigned?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 42
• Spring 2004, grade 8, item 25
• Spring 2006, grade 8, item 31

(8.1.b) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations. The student is expected to select and use appropriate forms of rational numbers to solve real-life problems including those involving proportional relationships.

Clarifying Activity with Assessment Connections

The teacher provides students with real-life situations that involve comparing numbers and computing costs. Students work together to determine solutions.

For example, Diane, Jim, and Cindy bought a large pizza with 16 slices. Diane ate 25% of the pizza, Jim ate 6 slices, and Cindy ate 1/8 of the pizza. Who ate the most pizza and how do you know? The cost of the pizza is $1.25 for every two slices. How much should each person pay? Whenever possible, the student should write a proportion and use a variable to represent an unknown. Example:

Assessment Connections

Questioning . . .

Open with . . .

• Describe the different ways you could choose to represent the portion of the pizza each person ate.

Probe further with . . .

• Why do you think it's appropriate to use the representations you selected?
• In what situations is it most appropriate to give a response in fractional form? Decimal form? As a percentage?
• How might other representations be used to solve the problem?
• How did you determine the cost of the pizza each person ate?

Listen for . . .

• Can the student justify the selected number form?
• Does the student use appropriate vocabulary for representing numbers in different forms?

Look for . . .

• Does the student use different number forms appropriately (fractions, percentages, decimals, proportions, etc.)?
• Does the student correctly convert numbers from one form to another?
• What strategy does the student use to convert numbers from one form to another?
• What methods does the student use to find the cost?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 45
• Spring 2004, grade 8, item 40
• Spring 2006, grade 8, item 27

(8.1.c) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations. The student is expected to approximate (mentally and with calculators) the value of irrational numbers as they arise from problem situations (such as π, ).

Clarifying Activity with Assessment Connections

The student is given a piece of 5 x 5 dot paper. [Download number cards here (pdf 4kb).] Students are asked to draw squares of every size possible (including tilted squares, there are 8 possible squares). A vertex of the square must be one of the dots. (The length of sides of possible squares on a 5 x 5 grid are: 1, , 2, , , 3, , and 4.) (Note: Students may want to explore first on a geoboard.)

The student examines the dimensions of each of the squares that are found. He or she determines the length of the sides using the Pythagorean Theorem when necessary. If the length of the side is an irrational number, the student mentally estimates an approximation to the number.

The student is given a number line with intervals that are equal to the intervals of the dot paper to check the mental estimates. The student approximates the length of each square by cutting out all of the possible squares and placing them on the number line to measure.

Students will verify their approximations of square roots using a calculator.

Assessment Connections

Questioning . . .

Open with . . .

• Describe the squares that you found.

Probe further with . . .

• Can you find more squares?
• What are some other ways you can draw a square in this area?
• Is there a difference between the tilted and non-tilted squares when you place them on the number line?
• Where do the tilted squares fit on the number line?
• How do you know if you have all the squares?

Listen for . . .

• Does the student verify the value of the square root to its placement on the number line?
• Can the student describe a systematic approach for finding all the squares?

Look for . . .

• Does the student include tilted squares?
• Is the student placing the numbers correctly on the number line?
• Does the student use a systematic method for finding all the squares?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 30
• Spring 2004, grade 8, item 42
• Spring 2006, grade 8, item 18

(8.1.d) Number, operation, and quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations. The student is expected to express numbers in scientific notation, including negative exponents, in appropriate problem situations.

Clarifying Activity with Assessment Connections

Students look through newspapers and magazines or their science textbooks for examples of very large or very small numbers. Students are then asked to express these numbers in scientific notation. The students then post their examples on a bulletin board. After all examples are displayed, the class compares and contrasts the numbers.

Assessment Connections

Questioning . . .

Open with . . .

• What did you do to write the numbers in scientific notation?

Probe further with . . .

• Do you see a way to group these numbers?
• What do some of these numbers have in common?
• How do you determine if these numbers are equivalent?
• What are the place values of specific digits?
• What is the purpose of the exponent?
• Why would you write some numbers in scientific notation?

Listen for . . .

• Can the student differentiate between very large and very small numbers?
• Does the student recognize various representations of numbers?

Look for . . .

• Are the numbers in scientific notation?
• Is the power of ten used correctly?
• Does the student recognize the difference between positive and negative exponents?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 46
• Spring 2004, grade 8, item 43
• Spring 2006, grade 8, item 13

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(8.2) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions.

(8.2.a) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions. The student is expected to select appropriate operations to solve problems involving rational numbers and justify the selections.

Clarifying Activity with Assessment Connections

Students create a restaurant menu and write problems that use addition, subtraction, multiplication, and/or division. For example, Hannah pays for her meal with a $20 bill. She ordered a salad for $3, a hamburger for $8, and two sodas for $1.50 each. If tax is 8.25% and she wants to leave a 15% tip, how much change should Hannah receive? After students have created a problem, have them switch problems with another student. Each student then creates an expression by selecting the appropriate operations that could be used to solve the problem. In addition, the student must provide a detailed justification for the selections made.

Assessment Connections

Questioning . . .

Open with . . .

• What strategy could you use to determine your bill?

Probe further with . . .

• Can you estimate your bill?
• How was the total generated?
• What different strategies can be used to calculate a tip?
• When should you calculate the tax? The tip? Does it make a difference?
• How do you calculate the amount of change received?

Listen for . . .

• Does the student verbalize the strategy using appropriate vocabulary?
• Does the student accurately describe a procedure to find the amount of change?
• Does the student know to add the tax and tip to the bill?
• Is the student using estimation skills to verify results?

Look for . . .

• What strategies did the student use?
• Is the student selecting the appropriate operations to create the expression?
• Is the student justifying his or her strategies?
• Does the student know how to rewrite the percentage as a decimal?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2006, grade 8, item 47

(8.2.b) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions. The student is expected to use appropriate operations to solve problems involving rational numbers in problem situations.

Clarifying Activity with Assessment Connections

Students are divided into groups. Each group is given a selection of local newspaper sale advertisements. The students make decisions about what to purchase given a certain amount of money to spend. Calculations should include actual prices, sale prices, discounts (percentages), taxes, and other factors. Each group shares their results with the class, including what purchases they decided to make and why.

Assessment Connections

Questioning . . .

Open with . . .

• Explain your strategy for determining the choices and calculations you made.

Probe further with . . .

• What strategies can you use to determine the amount of your purchases?
• What did you purchase with your money?
• How much money did you spend?
• How much money did you have left?
• How would a discount affect your purchases?

Listen for . . .

• Can the student verbalize the strategies used?
• How does the student decide what to spend?
• Can the student describe the process correctly for finding the discount? Tax? Final price?

Look for . . .

• Does the student perform the calculations correctly?
• Does the student justify his or her decisions?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 21
• Spring 2004, grade 8, item 35
• Spring 2006, grade 8, item 11

(8.2.c) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions. The student is expected to evaluate a solution for reasonableness.

Clarifying Activity with Assessment Connections

Students consider the following situation:

In the first half of the baseball season, Stan gets 65 hits out of 253 times at bat. In the second half of the season, Stan gets 70 hits out of 284 times at bat. What is his batting average for the entire season?

Students discuss whether the sum of his batting averages for the two halves of the season (0.257 + 0.246 = 0.503) is a reasonable answer to the question.

(Since a batting average is the ratio of hits to times at bat, 0.503 means that the player hit the ball over half of the times at bat, so it is not a reasonable answer. Stan's batting average for the entire season is (65 + 70)/(253 + 284).)

Assessment Connections

Questioning . . .

Open with . . .

• What do the numbers 0.257, 0.246, and 0.503 mean?

Probe further with . . .

• If Stan's batting average were 0.503, what does that mean about how he performed?
• How were 0.257 and 0.246 computed?
• What is Stan's batting average for the season?
• If Stan were at bat 10 times in one game, how many hits would you expect him to have?
• Is the sum method of calculating Stan's average for the entire season reasonable?
• Why do you get different answers?

Listen for . . .

• Is the student able to correctly explain the meaning of batting average?
• Does the student understand the difference between adding the two ratios and computing the ratio for the entire season?

Look for . . .

• Does the student use the decimal representation correctly?
• How does the student compute the ratio and divide and round to get the decimal value?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 26
• Spring 2004, grade 8, item 16
• Spring 2006, grade 8, item 43

(8.2.d) Number, operation, and quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions. The student is expected to use multiplication by a constant factor (unit rate) to represent proportional relationships.

Clarifying Activity with Assessment Connections

Students use the scale of a map to calculate the relationship between the distance on the map and the actual distance between two points. For example, if a scale on a map reads that 1 inch = 3 miles, then the number of miles between the two points (real distance) is three times the number of inches on the map (map distance), or real distance = 3(map distance). Students then use this relationship to find real distances between given points on the map, or to extend the map further while maintaining the same scale.

Assessment Connections

Questioning . . .

Open with . . .

• How did you use the scale on the map to determine the actual distances?

Probe further with . . .

• What is the scale on the map?
• What does this represent?
• Is there more than one way to use the scale to find the actual distance?
• How can you use the scale to determine actual distance?

Listen for . . .

• Does the student verbalize the relationship between the map scale and actual distance?
• Does the student make a generalization about finding distance?

Look for . . .

• Does the student correctly determine the distances?
• Does the student always refer to the original scale or does he or she use an equivalent form to determine values?
• Does the student use the constant factor (the scale) to find distance?
• What operations does the student use?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 37
• Spring 2006, grade 8, item 40

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(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems.

(8.3.a) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems. The student is expected to compare and contrast proportional and non-proportional linear relationships.

Clarifying Activity with Assessment Connections

The teacher puts students into small groups and gives them the following scenario:

The student council is planning a trip to a water park. There are two parks in the area, Splish-Splash and Waterwonders. Assume that both parks are equal in popularity. Splish-Splash charges $12.50 per student, while Waterwonders charges $9.50 per student plus a $100.00 group fee for any size group. Which park would be the most economical?

Create a convincing presentation to the president of the student council, pointing out which park is a better deal and why. Support your presentation with tables, charts, graphs, and equations.

Assessment Connections

Questioning . . .

Open with . . .

• Which park is a better deal?

Probe further with . . .

• Does it matter how many students are in the class?
• How much is it for 10 students to attend Splish-Splash? Waterwonders?
• How much is it for 20 students to attend Splish-Splash? Waterwonders?
• If the number of students is doubled, does the price also double at both parks? Why or why not?
• For how many students would the cost be equal at both parks?
• How can you tell from your table and your graph when the cost is equal for both parks?
• Does one park always offer the better deal?
• Is the pricing proportional at either park?
• For which park is the pricing proportional? If one of these parks has proportional pricing and one does not, which is proportional and how do you know?
• Can you use the graphing calculator to help?
• How does the proportionality (or lack of it) appear in your table, graph, or equation?

Listen for . . .

• Is the student able to articulate that the better deal depends on the number of students going?
• Does the student articulate the difference between proportional and non-proportional relationships in tables, graphs, and equations?
• Can the student verbalize and interpret an equation for each situation?
• Can the student present the information in multiple ways?

Look for . . .

• Is the student recognizing that one situation is proportional and one is not?
• Is the student generating tables, graphs, or equations?
• Does the student see the intersection of the two situations in the table?
• Does the student see the intersection of the two situations in the graph?
• Does the student use the correct y-intercept?
• Does the student recognize the relationship between the table, the graph, and the equations?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, items 32 and 49
• Spring 2006, grade 8, item 49

(8.3.b) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems. The student is expected to estimate and find solutions to application problems involving percents and other proportional relationships such as similarity and rates.

Clarifying Activity with Assessment Connections

In the schoolyard students measure the height of other students and the length of their shadows. Using this information, students estimate and then calculate the height of a building or any other structure in the schoolyard. Students are then instructed to draw a picture showing the measurements taken and to represent the information in a table and a graph.

Assessment Connections

Questioning . . .

Open with . . .

• How did you determine the height of a structure in the schoolyard?

Probe further with . . .

• What is the relationship between the structures' heights and the shadows' lengths?
• What other measurements do you need?
• Can you use this relationship to determine the height of other objects around the school?
• What information do you need to graph the information?

Listen for . . .

• Does the student understand the relationship between the two shadows?
• Does the student connect this activity to similar triangles?
• Can the student describe the similar triangles for the situations?
• Does the student use appropriate language to express the relationships between the objects and their shadows?
• Does the student generate an equation relating to the object height and its shadow?

Look for . . .

• Does the student use appropriate estimation strategies?
• Does the student use correct measurement techniques?
• Does the student use appropriate units of measure?
• Can the student draw and appropriately label figures for this situation?
• Does the student graph the information appropriately?
• Is the student using appropriate numbers and operations to solve the problem?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 21
• Spring 2006, grade 8, items 14 and 28

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(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship.

(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship. The student is expected to generate a different representation of data given another representation of data (such as a table, graph, equation, or verbal description).

Clarifying Activity with Assessment Connections

Students collect data on the amount of time a person can hold his or her breath and the diameter of a balloon that person can fill with one continuous breath. Comparing the two measurements, students display the data in a table and then graph the relationship between the two experiments.

Assessment Connections

Questioning . . .

Open with . . .

• Which representation shows a better relationship between the two variables?

Probe further with . . .

• Once distance around the balloon is measured and recorded, how do you find the diameter of the balloon?
• Which pieces of data are relevant?
• What are some ways to represent this data?
• Should the amount of time each individual can hold their breath be recorded with the diameter of their balloon, or does this matter?
• Which should come first, time of breath held or balloon diameter? Does it matter?
• Can this data be represented in a table? In a graph? In an equation? In writing? Verbally?
• What kind of graph best represents this data? Why?
• Can you use the graphing calculator to represent your data? In how many ways can you do this?
• Which representations of your data help you draw conclusions or make generalizations about your experiments?
• Is there a pattern in this relationship that can be described using symbols? Why or why not?
• How does the representation differ if we use circumference instead of diameter? Radius instead of diameter?

Listen for . . .

• Is the student using appropriate vocabulary to describe the data (i.e., diameter, units of time)? Does the student use words like table, ordered pair, axes, graph, plot, and so on when discussing representations?
• How does the student verbally describe the numerical relationship between length of time the breath was held and the diameter of the balloon? (For example, "As one increases, the other increases.")
• Can the student verbally describe the data? (For example, "These numbers are the times recorded for students holding their breaths, and the other numbers are the lengths of the diameter.")
• Can the student verbally describe the connections between two or more representations? (For example, "The column in this table is represented by the axis in this graph or this ordered pair is located here on the graph.")

Look for . . .

• Does the student have an organizational device to record the data?
• Can the student self-monitor and self-correct?
• Does the student check for reasonable units of measure (time and length) in the representations and the collected data?
• Can the student organize the collected data (relate length of time and the diameter of the balloon to same person)?
• Does the student accurately represent the data?
• Can the student generate different representations of the data collected?
• Does the student label the data correctly?
• Does the student use the graphing calculator to represent the data in various ways? Can the student use the graphing calculator to create lists, tables, scatter plots, etc.?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, items 1 and 47
• Spring 2004, grade 8, item 7
• Spring 2006, grade 8, items 3 and 32

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(8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems.

(8.5.a) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems. The student is expected to predict, find, and justify solutions to application problems using appropriate tables, graphs, and algebraic equations.

Clarifying Activity with Assessment Connections

Using the data in 8.4, students make predictions about other (non-observed) breath times and the related inflated balloon diameters.

Assessment Connections

Questioning . . .

Open with . . .

• What relationship exists between the amount of time a person can hold his or her breath and the diameter of a balloon the same person inflates with a continuous breath?

Probe further with . . .

• Is there a trend to this data?
• Which representations of your data help you draw conclusions or make generalizations?
• What factors affect your data (e.g., ceiling effect of the size of the balloon)?
• What is the amount of time breath is held for the balloon with the largest diameter? With the smallest diameter? Look at another set of data; does the time and size follow the trend?
• As the amount of time breath is held increases, what happens to the balloon diameter?
• As the diameter decreases, what happens to the amount of time breath is held?
• Use the trend observed in the data to predict how long a person can hold their breath if they can fill a balloon to a diameter of 5 inches in one continuous breath.
• If a whale can hold its breath for 4 1/2 minutes, what size balloon could it inflate?
• Is there data that doesn't fit the trend? What may cause this?
• What is the relationship between the amount of time the breath can be held and the diameter of the balloon? Why?
• Is there a pattern in this relationship that can be described using symbols? Why or why not?
• How can you use your graphing calculator to represent the data?

Listen for . . .

• Can the student interpret the data and make generalizations?
• Can the student identify trends in the data?
• Can the student correctly make predictions based on trends in the data?
• Does the student use appropriate vocabulary (relationship, trend, prediction, table, graph, plot, or scatter plot)?
• Does the student use different representations to justify conclusions about the relationship between the data?
• Can the student identify independent and dependent variables?

Look for . . .

• Can the student write a justification for his or her prediction?
• Can the student correctly make predictions based on the trends in the data?
• Can the student use the graphing calculator to create lists, tables, scatter plots, etc.?
• Can the student evaluate reasonableness of predictions made?
• Does the student self-monitor and self-correct errors?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2006, grade 8, items 21, 38, and 46

(8.5.b) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems. The student is expected to find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change).

Clarifying Activity with Assessment Connections

The teacher gives students the following scenario:

A tile layer created these patterns:

Can you write a rule that will help the tile layer find the number of tiles needed for the nth figure in order to continue this pattern?

Assessment Connections

Questioning . . .

Open with . . .

• How will you determine how many tiles will be needed for the fifth figure? The sixth figure? The twelfth figure? The 100th figure?

Probe further with . . .

• If you extend this pattern to its 100th figure, how many tiles do you need?
• Using the figures given, how will you determine the 5th figure? The 10th figure? The 100th figure?
• What patterns do you see?
• Do you need to draw every figure from figure 4 to figure 100? Why not?
• What are some more efficient ways than drawing?
• Can you describe the pattern in words?
• Can you describe the pattern in symbols?
• Can you write a rule to describe this pattern?
• What is the relationship between the figure number and the number of tiles needed?
• Can you use this relationship to help you find the number of tiles needed for the fourth and fifth figures?
• What generalizations can you make?
• If the figure has 29 tiles, what figure number is it?
• If the figure has 229 tiles, what figure number is it?
• Can you use a graphing calculator to test your rule? How?

Listen for . . .

• Does the student discuss how the pattern continues?
• Can the student verbalize the rule?
• Does the student check his or her work for reasonableness? With and without the graphing calculator?
• Can the student correctly make predictions based on the rule?
• Does the student use appropriate vocabulary?
• Does the student justify the rule?
• Can the student differentiate the term number from the number of tiles?

Look for . . .

• Does the student organize information? Can the student use a systematic process to develop the rule?
• Does the student draw additional figures?
• Does the student use symbols to describe the pattern or rule?
• Does the student make a table?
• Does the student recognize flaws or errors in the rule and adjust it accordingly?
• Does the student use the calculator appropriately?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 32
• Spring 2006, grade 8, items 10 and 24

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(8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense.

(8.6.a) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense. The student is expected to generate similar figures using dilations including enlargements and reductions.

Clarifying Activity with Assessment Connections

Working in small groups, students lay a 1-centimeter grid transparency onto simple pictures and use the relationship of the picture to the squares on the grid (the properties of dilation) to draw a similar picture onto a larger grid, making an enlargement of the picture. (Note: If transparencies are not available, pictures may be drawn or copied directly onto a grid.)

Each group can use a different size grid for its enlargement, compare their enlargements, and use measurements of corresponding parts to identify the scale factor for each enlargement.

Assessment Connections

Questioning . . .

Open with . . .

• How is your enlargement like your original and how is it different?

Probe further with . . .

• How do you determine corresponding parts of figures?
• How could you describe the relationship between the corresponding sides of the two figures?
• How could you describe the relationship between the corresponding angles of the two figures?
• How much did the measurements of the corresponding parts of your new figure increase when compared to the original figure?
• How does your previous knowledge of similarity relate to this activity?

Listen for . . .

• Can the student use geometric terms such as dilation, enlargement, scale factor, growth, or multiples to describe the change in size?
• Does the student verbalize strategies about drawing the enlargement?
• Does the student formulate questions to problem-solve during the activity?
• Can the student use geometric terms to describe the similarity between the original and its enlargement?
• Does the student formulate questions to analyze the changes in the figure?
• Does the student interact with group members to formulate relationships and establish patterns among the figures?

Look for . . .

• Can the student accurately copy the original figure to the transparency grid?
• Can the student recognize if the enlargement is an accurate duplicate of the original figure?
• Does the student self-check the enlargement for an accurate scale factor?
• Does the student demonstrate strategies to find the scale factor of the dilation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 31
• Spring 2004, grade 8, item 15
• Spring 2006, grade 8, item 36

(8.6.b) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense. The student is expected to graph dilations, reflections, and translations on a coordinate plane.

Clarifying Activity with Assessment Connections

Students dilate, reflect, or translate a polygon on a coordinate plane, compare the coordinates of the new shape to the corresponding coordinates of the original shape, and look for patterns in the pairs of coordinates that describe the transformation.

Example: Reflect the figure over the x-axis. Give the new coordinates. Reflect the original figure over the y-axis. Give the new coordinates.

Assessment Connections

Questioning . . .

Open with . . .

• What patterns do you notice in the ordered pairs from the original figure and the way they correspond to the ordered pairs of the transformation?

Probe further with . . .

• After the figure reflects over the x-axis, what do you notice about the values of the x-coordinates in the original figure compared to the values of the x-coordinates of the transformed figure?
• After the figure reflects over the y-axis, what do you notice about the values of the y-coordinates in the original figure compared to the values of the y-coordinates of the transformed figure?
• What do you notice about the y-coordinates of the two figures?

Listen for . . .

• Can the student verbalize patterns found in the corresponding coordinates?
• Can the student describe the differences between the processes for each type of transformation?
• Can the student use the appropriate vocabulary when identifying transformations?

Look for . . .

• Can the student plot the points correctly?
• Can the student create a coordinate plane with correct symbols?
• Can the student make the connection from transforming points visually one by one to making a generalization and applying the generalization to the set of coordinates?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, items 10 and 43
• Spring 2004, grade 8, item 27
• Spring 2006, grade 8, item 7

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(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.

(8.7.a) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to draw three-dimensional figures from different perspectives.

Clarifying Activity with Assessment Connections

Groups of students build three-dimensional models with blocks and then sketch the top, front, and side views. Groups trade their sketches with another group and ask them to build a model from the sketches.

For additional activities visit the National Council of Teachers of Mathematics Illuminations website at illuminations.nctm.org.

Assessment Connections

Questioning . . .

Open with . . .

• Justify how your sketches are congruent to the model. Justify how your model is congruent to the sketches.

Probe further with . . .

• What does the front of the model look like? The side? The top? The back?
• Is there more than one model that could be built using the sketches?
• Are there other views to sketch from the model?
• Given other views such as the bottom, back, or left view, is the model still the same?

Listen for . . .

• Does the student use vocabulary to describe position while building and sketching the models?
• Does the student verify number of cubes to support accuracy of models and sketches?

Look for . . .

• Can the student construct a model?
• Can the student sketch the model?
• Does the model match the sketches?
• Can the student construct different perspectives of the same model?
• Can the student sketch different perspectives of the same model?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 50
• Spring 2004, grade 8, item 44
• Spring 2006, grade 8, item 50

(8.7.b) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to use geometric concepts and properties to solve problems in fields such as art and architecture.

Clarifying Activity with Assessment Connections

Students design kites on paper, using similar and congruent geometric shapes. Students then create a scale drawing and determine the perimeter and area of the scale drawing and the kite. (Teachers may want to include some minimum requirements for the kites—for example, use of at least three different shapes.)

For additional activities visit the National Council of Teachers of Mathematics Illuminations website at illuminations.nctm.org.

Assessment Connections

Questioning . . .

Open with . . .

• What geometric shapes did you use in creating your kite?

Probe further with . . .

• Which ones are similar and/or congruent?
• How do you know your shapes are similar?
• What can you determine about the relationship between the lengths of the sides of the similar or congruent figures? Describe the relationships between the perimeters of the congruent shapes.
• Describe the relationships between the areas of the similar shapes.
• Explain what strategies you used to determine the relationships between the shapes.
• Describe the relationship between the angles of similar shapes.

Listen for . . .

• Can the student express the various relationships between similar or congruent shapes?
• Is the student using the appropriate vocabulary?
• Is the student using geometric properties to describe the figure?

Look for . . .

• Is the student demonstrating an understanding of scale factor in similar figures?
• What geometric relationships does the student use to draw the scale drawing?
• Does the student recognize the relationship between the area of the scale model and the area of the actual kite?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2006, grade 8, item 30

(8.7.c) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to use pictures or models to demonstrate the Pythagorean Theorem.

Clarifying Activity with Assessment Connections

Students are given grid paper, the table below, and two drawings. Students are then instructed to create the drawings for the other rows of the table, complete the table, and look for a relationship among the areas of squares drawn on lengths of the sides of right triangles.

Drawing one represents the first row of the table.

Drawing two represents the last row of the table.

For additional activities visit the National Council of Teachers of Mathematics Illuminations website at illuminations.nctm.org.

Assessment Connections

Questioning . . .

Open with . . .

• What is the relationship between the areas of the squares and the lengths of the sides?

Probe further with . . .

• Is there a relationship between the areas of the squares on leg 1, leg 2, and the hypotenuse?
• What patterns do you see in the rows of your table?
• Can you write a general rule for the patterns you see?
• How do you find the areas of the squares?
• What is the relationship between length of the sides of a square and its area?
• Could you draw models of examples other than those in the table?
• How could you find the lengths of the sides of the square without using a ruler?

Listen for . . .

• Is the student making conjectures about the patterns he or she recognizes?
• Does the student see a pattern among the areas of the three squares?
• Does the student identify the relationship between the area of the square on the hypotenuse and the length of the square's side?
• Does the student know how to find area?

Look for . . .

• What is the student doing to estimate the area of the square on the hypotenuse?
• Does the student complete the table correctly?
• Does the student find the areas correctly?
• Does the student communicate the patterns he or she recognizes?
• Does the student recognize a relationship between the pictures and the table?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 37
• Spring 2004, grade 8, item 22

(8.7.d) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to locate and name points on a coordinate plane using ordered pairs of rational numbers.

Clarifying Activity with Assessment Connections

Students create a design on a coordinate plane, including points in all four quadrants and points with coordinates other than integers. Students exchange designs. The students select a starting point on the design and list the ordered pairs in the order they occur while tracing the design. The students exchange the lists of ordered pairs and plot the points to see if the design is created by the list.

Assessment Connections

Questioning . . .

Open with . . .

• How do you locate and name ordered pairs of rational numbers on a coordinate plane?

Probe further with . . .

• What is an ordered pair?
• How are points in the different quadrants alike? Different?
• What is unique about the ordered pairs in quadrant I? II? III? IV?
• What does the first number of an ordered pair tell you?
• What does the second number of an ordered pair tell you?
• How do you locate ordered pairs that contain rational numbers?

Listen for . . .

• Does the student estimate the points in the coordinate plane correctly?
• Does the student recognize the difference in ordered pairs among the different quadrants?
• Does the student approximate the location of ordered pairs with rational numbers correctly?

Look for . . .

• Does the student locate a point that has rational numbers as its ordered pairs?
• Does the student label the coordinate plane correctly?
• Does the student plot the ordered pairs correctly?
• Does the student self-monitor and self-correct as he or she is plotting the points?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 2
• Spring 2006, grade 8, items 5 and 35

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(8.8) Measurement. The student uses procedures to determine measures of three-dimensional figures.

(8.8.a) Measurement. The student uses procedures to determine measures of three-dimensional figures. The student is expected to find lateral and total surface area of prisms, pyramids, and cylinders using concrete models and nets (two-dimensional models).

Clarifying Activity with Assessment Connections

Students measure the dimensions of prisms, pyramids, and/or cylinders and then use grid paper to make nets for the three-dimensional figures. Students cut out the net and verify that it actually covers the object. They then count the squares on the grid paper to estimate the surface areas of the objects.

For additional activities visit the National Council of Teachers of Mathematics Illuminations website at illuminations.nctm.org.

Assessment Connections

Questioning . . .

Open with . . .

• How can you describe the net needed to represent the surface area of a cylinder, pyramid, and prism?

Probe further with . . .

• Describe the shapes of the bases and lateral area of each three-dimensional figure.
• What strategies could you use to count the squares on the grid paper?
• How could you find the surface area without the grid paper?
• How are the dimensions of the lateral surface related to the dimensions of the bases?

Listen for . . .

• Does the student recognize that the bases of a cylinder and prism are congruent?
• Does the student recognize that the lateral area of a cylinder is a rectangle?
• Does the student connect the length of the rectangle to the circumference of the circular bases?
• Does the student recognize that the surface area of a cylinder can be calculated by using the formulas for the area of a circle and a rectangle?

Look for . . .

• Does the student draw the net correctly?
• Is the student using efficient methods to count the squares on the grid paper?
• Does the student correctly transfer the measurements of the three-dimensional figure to the net?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 13
• Spring 2006, grade 8, item 22

(8.8.b) Measurement. The student uses procedures to determine measures of three-dimensional figures. The student is expected to connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects.

Clarifying Activity with Assessment Connections

Students are given rectangular prisms and asked to determine the area of the base by covering the bottom with cubes. Layering the cubes to match the first layer on the base, students fill the prism to determine the volume of the prism. Students connect the procedure (finding the area of the base x the number of layers) to the formula for calculating the volume of a rectangular prism [area of the base x height, or (l x w of the base) x height].

Assessment Connections

Questioning . . .

Open with . . .

• What strategies could you use to find the volume of the prism? How did you determine your answer?

Probe further with . . .

• Approximately how many cubes will it take to fill one layer?
• Approximately how many layers will it take to fill the prism?
• Approximately how many cubes will it take to fill the prism?
• How is the process of finding the volume of the cube related to the units for volume?

Listen for . . .

• Is the student first determining the number of cubes on one layer and then multiplying by the number of layers?
• Does the student verbalize connection between the number of cubes on one layer and "length x width"?
• Does the student connect the number of layers to the height of the prism?
• Does the student make the connection between the number of cubes needed to fill the prism and the product of "length x width x height"?

Look for . . .

• Is the student completely filling the prism with cubes to determine the volume or is the student filling a row representing the length, a column representing the width, and a tower representing the height, then calculating the volume?
• What strategies does the student use to look for patterns (for example, making a table, generalizing from the model, etc.)?
• Can the student generalize a formula to find volume?
• Does the student recognize that orientation does not affect the volume?

TAKS Connection

• This student expectation is not tested on TAKS. Although this is not directly tested at Grade 8, it is an important foundation for student expectations tested at later grades.

(8.8.c) Measurement. The student uses procedures to determine measures of three-dimensional figures. The student is expected to estimate measurements and use formulas to solve application problems involving lateral and total surface area and volume.

Clarifying Activity with Assessment Connections

Students are given the dimensions of a hollow cylinder, such as a piece of pipe:

Students are asked to find the total surface area of the cylinder, including the interior, and to find what volume would be needed to fill the interior of the hollow cylinder. Students then apply the procedure to finding the surface area of hollow cylinders in general.

(Possible strategy for finding the surface area: The surface area of the hollow cylinder = the surface area of the large cylinder + the lateral surface area of the small cylinder - the areas of the two bases of the small cylinder. If h is the height of both cylinders, r is the radius of the large circle, and s is the radius of the small circle, then the surface area of the hollow cylinder = .

Assessment Connections

Questioning . . .

Open with . . .

• What strategies could you use to determine the total surface area of the hollow cylinder? Volume?

Probe further with . . .

• If it were possible to cut the pipe along its length and unwrap it, could you draw this shape?
• Have you considered the interior of the pipe?
• What is the relationship between the interior and exterior surface area?
• How many square feet would it take to cover the lateral area of the outside pipe?
• How many square feet would it take to cover the lateral area of the inside of the pipe?
• What do the ends of the pipe look like?
• How is the surface area of the ring at the top and bottom of the pipe related to the inside and outside cylinders? How could you determine its surface area?
• What are the three quantities that could vary in this situation to get the formula for the surface area of any pipe?
• How would you write a general formula for the surface area of a hollow pipe?
• How could you estimate the volume of the pipe?
• What measurements will help you find the volume of the pipe?

Listen for . . .

• Does the student include the small cylinder when describing the surface area of the pipe?
• Is the student using appropriate formulas to describe the lateral surface area of a cylinder and area of a circle?
• Does the student use the appropriate geometric vocabulary?
• Does the student verbalize the relationship that exists between the larger and smaller circles on the pipe ends?
• Does the student find the volume of the interior of the pipe?
• Does the student recognize that there is difference in the volume because of the thickness of the pipe?
• Does the student see a relationship between finding volume and surface area?

Look for . . .

• Is the student using formulas correctly when finding surface area? Volume?
• Is the student using appropriate formulas to determine lateral surface area of a cylinder and area of a circle?
• Does the student use appropriate units?
• Can the student use a systematic approach to develop the general formula?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 28
• Spring 2004, grade 8, item 33
• Spring 2006, grade 8, item 2

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(8.9) Measurement. The student uses indirect measurement to solve problems.

(8.9.a) Measurement. The student uses indirect measurement to solve problems. The student is expected to use the Pythagorean Theorem to solve real-life problems.

Clarifying Activity with Assessment Connections

The student is given a diagram showing the street plan of a city on a grid. A taxi travels on the street between two locations and the helicopter flies directly from one location to the other. The student uses the Pythagorean Theorem to compare the distances traveled by the taxi and the helicopter.

Assessment Connections

Questioning . . .

Open with . . .

• How are the distances traveled by the taxi and the helicopter alike or different?

Probe further with . . .

• How can you find the distance the taxi traveled?
• Is there more than one way to find the distance traveled by the taxi?
• How can you find the distance the helicopter traveled?
• Between what two integer values does the distance the helicopter traveled lie?
• Using your calculator, can you find an approximation for this distance?

Listen for . . .

• Does the student use appropriate vocabulary?
• Does the student correctly describe the use of the Pythagorean Theorem?
• Does the student explain the process for estimating the square root?

Look for . . .

• Is the student correctly counting the distances on the grid?
• Is the student correctly using the Pythagorean Theorem?
• Does the student demonstrate an understanding of square root?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 19
• Spring 2006, grade 8, items 33 and 45

(8.9.b) Measurement. The student uses indirect measurement to solve problems. The student is expected to use proportional relationships in similar two-dimensional figures or similar three-dimensional figures to find missing measurements.

Clarifying Activity with Assessment Connections

In the schoolyard, students identify an object, such as a flagpole, that they would like to find the height of. Then students measure the height of another student, the length of the other student's shadow, and the length of the selected object's shadow. Using what they know about similar triangles and proportional relationships, students find the height of the selected object. The teacher asks students to convert their measurements within the same system of choice (i.e., inches to feet, centimeters to millimeters, etc.)

Assessment Connections

Questioning . . .

Open with . . .

• What is the relationship between the heights and the shadows?

Probe further with . . .

• How are these measurements related? How do you know?
• How do you describe their relationship?
• How do you determine an unknown height?
• Is there more than one way to determine a height?

Listen for . . .

• Does the student use correct units in describing the measurements?
• Does the student correctly express the relationship between the measurements?
• Does the student correctly communicate how to determine an unknown height?
• If the student converted the measurements within the same system, was the conversion performed correctly?

Look for . . .

• Does the student accurately measure the lengths?
• How does the student organize the information collected?
• Does the student correctly represent the proportional relationships?
• Does the student recognize the constant of proportionality?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 7
• Spring 2004, grade 8, item 36
• Spring 2006, grade 8, item 17

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(8.10) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures.

(8.10.a) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures. The student is expected to describe the resulting effects on perimeter and area when dimensions of a shape are changed proportionally.

Clarifying Activity with Assessment Connections

Given the following grid with shapes that have proportionally changed lengths and widths, students find the perimeter and area for each shape and organize the data in a table. Students are then instructed to look for and list patterns that describe the relationship between the changes in the shapes' dimensions and the changes in the shapes' perimeters and areas (e.g., halving the length and width of a rectangle gives 1/2 of its original perimeter, but gives only 1/4 as much area).

Assessment Connections

Questioning . . .

Open with . . .

• What is the relationship between the perimeter and area of these figures?

Probe further with . . .

• Can you use a formula to find the perimeter?
• Can you use a formula to find the area?
• What is the relationship between corresponding sides of any two figures in the grid?
• What is the relationship between the areas of any two figures in the grid?
• What is the relationship between the perimeters of any two figures in the grid?
• If you draw a figure with sides that are four times the length of the sides of figure A, what would you expect the perimeter of the new figure to be?
• If you draw a figure with sides that are four times the length of the sides of figure A, what would you expect the area of the new figure to be?

Listen for . . .

• Does the student recognize patterns in the table?
• Does the student communicate using the properties of proportionality?
• Does the student correctly describe perimeter and area?
• Does the student express an understanding that perimeter increases by the scale factor in similar figures?
• Does the student express an understanding that area increases by the scale factor squared in similar figures?

Look for . . .

• Can the student organize a table?
• How does the student organize the information?
• Does the student correctly compute the perimeter and area?
• Does the student use appropriate formulas to find the perimeter and area?
• Does the student distinguish between the perimeter units and the area units of measure?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 28

(8.10.b) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures. The student is expected to describe the resulting effect on volume when dimensions of a solid are changed proportionally.

Clarifying Activity with Assessment Connections

Given diagrams of solids with proportionally changed dimensions, students find the volume for each solid and organize the data in a table. Students are instructed to look for patterns that describe the relationship between changes in the dimensions of the solids and changes in the volumes of the solids (e.g., doubling the length, width, and height of a rectangular prism gives eight times the volume).

Assessment Connections

Questioning . . .

Open with . . .

• What is the relationship between the dimensions of the solid figures and their volumes?

Probe further with . . .

• How can you find the volume of the figures? Could you use a formula to find the volume?
• What is the relationship between the height and diameter of any two figures?
• What is the relationship between the volumes of any two figures?
• If you draw a figure with height and diameter six times the height and diameter of figure A, what would you expect the volume of the new figure to be?

Listen for . . .

• Does the student exhibit an understanding of the concept of volume?
• Does the student correctly convey how to determine volumes?
• Does the student use appropriate mathematical vocabulary?
• Does the student express an understanding that volume increases by the scale factor cubed in similar figures?
• Does the student communicate using the properties of proportionality?

Look for . . .

• Does the student recognize that the diameter was given and he or she must determine the radius?
• How does the student organize the information?
• Does the student use the appropriate formula to find volume?
• Does the student correctly compute the volume?
• Does the student recognize that there is a proportional relationship between the heights of the solids and the radii of the given solids?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 38

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(8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions.

(8.11.a) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. The student is expected to find the probabilities of dependent and independent events.

Clarifying Activity with Assessment Connections

Students conduct experiments, collect data, and construct the sample spaces to find the experimental and theoretical probabilities of events resulting from:

• drawing cubes from a bag with replacement (independent events); and
• drawing cubes from a bag without replacement (dependent events).

Assessment Connections

Questioning . . .

Open with . . .

• How does the probability of drawing a red without replacement compare with the probability of drawing a red with replacement?

Probe further with . . .

• If we draw out two cubes, what are the possible outcomes?
• How do the probabilities compare for events conducted with replacement or without replacement?
• Are the outcomes equally likely or will one outcome occur more often than the others? Why?

Listen for . . .

• Is the student verbalizing that drawing green first and then red is different from drawing red and then green?
• Is the student using correct vocabulary?
• Is the student able to articulate the difference in probabilities for events conducted with replacement or without replacement?

Look for . . .

• How is the student using models to determine theoretical probability (for example, organized list, tree diagram, matrix)?
• Is the student able to record and organize his or her data in an appropriate way?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 22
• Spring 2004, grade 8, item 30
• Spring 2006, grade 8, item 37

Extension Using Technology

Students can use computer software to simulate a large number of trials and make generalizations and conjectures about the relationship between experimental and theoretical probabilities. They can also use computer software to graph this relationship.

For additional activities involving graphical representation of data, visit the National Council of Teachers of Mathematics Illuminations website at illuminations.nctm.org.

(8.11.b) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. The student is expected to use theoretical probabilities and experimental results to make predictions and decisions.

Clarifying Activity with Assessment Connections

Students may design a probability game—or use the two games described below. They investigate whether a game is fair by using experimental results and theoretical probabilities. Here, the game is considered to be fair if winning and losing are equally likely.

Game 1: Toss three two-color counters. You win if there are at least two red chips showing. Otherwise, you lose. Is the game fair? Why or why not?

Game 2: Roll two number cubes. If the product of the two numbers is even, you win. If it is odd, you lose. Is the game fair? Why or why not?

For each game, the students collect data by playing the game and calculate the experimental probability of winning after playing 10, 20, and 50 times. As a class, graph the results for playing 10 times on one graph, playing 20 times on another graph, and playing 50 times on a third graph.

Students then use this information to predict outcomes.

For example, if you played Game 1 100 times, about how many times would you expect to win? Game 2?

Assessment Connections

Questioning . . .

Open with . . .

• Tell me about your probability of winning this game.

Probe further with . . .

• Is the game fair?
• When you played this game 10 (20, 50) times, what was the experimental probability of you winning?
• How many times did you play? How many times did you win?
• What is the theoretical probability of you winning this game? How did you figure this out?
• What are the strategies you have used to find theoretical probabilities?
• How do the theoretical and experimental probabilities compare?
• When you graphed the experimental probability for 10, 20, and 50 games for the class, how did these experimental probabilities compare as the number of games increased?
• How does the experimental probability compare to the theoretical probability as the number of games played increases? (The experimental probability is more likely to be close to the theoretical probability.)

Listen for . . .

• Is the student able to analyze the data to make predictions?
• Does the student see the relationship between experimental and theoretical probabilities?
• Is the student able to justify ideas about the relationship between experimental and theoretical probabilities?
• Does the student analyze the game by examining whether or not the events are equally likely?
• Does the student recognize that the more experimental trials are conducted, the closer the result becomes to the theoretical probability?

Look for . . .

• Does the student's graph demonstrate that the experimental probability approaches the theoretical probability as the number of games played increases?
• Does the student have an organized method for making predictions (making a table, writing a proportion equation, using a scale factor, etc.)?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 8
• Spring 2004, grade 8, item 49
• Spring 2006, grade 8, item 41

(8.11.c) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. The student is expected to select and use different models to simulate an event.

Clarifying Activity with Assessment Connections

Given that the probability of getting a broken CD player from Bad CD Players, Inc., is 1/3, students select and use a model to simulate buying a CD player. For example, students could place two blue marbles (representing good CD players) and one red marble (representing a defective CD player) in a bag and draw one; or they could spin a spinner that is 1/3 red and 2/3 blue, with red representing a defective CD player. Students use the model to simulate the event and then analyze their data.

Assessment Connections

Questioning . . .

Open with . . .

• How can we create a model to simulate a particular event?

Probe further with . . .

• What are some models we have used in the past for finding probability (number cubes, colored tiles and blocks, spinners, coins, marbles, cards, calculators)?
• What does your data indicate about this event?
• Can you use your sample results to make predictions about the number of broken CD players in a large shipment?
• How would your results have been different if your sample size had been smaller? Larger?

Listen for . . .

• Is the student able to justify his or her choice of model?
• Is the student able to make connections between the models and the event?
• Is the student able to make predictions for a larger sample?

Look for . . .

• Is the student able to use more than one model to simulate the event?
• Is the student recording the data in an organized way?

TAKS Connection

• This student expectation is not tested on TAKS. Although this is not directly tested at Grade 8, it is an important foundation for student expectations tested at later grades.

Extension Using Technology

• Students design random simulations to use with technology for the games created in Activity 8.11B.

  For game 1, students can program the Random Number Generator on a calculator to produce random numbers from 200 to 999. In this range, there are an equal number of digits in each place-value position that are even or odd; in other words, the probability of getting an even digit in any one position is 1/2, and the probability of getting an odd digit in any one position is 1/2. If an even digit represents red, and an odd digit represents yellow, then when the calculator's RNG shows 356, it is the same as tossing three two-color counters and getting one red and two yellows.

  For game 2, students can program the calculator's RNG to choose a random number from 1 through 6, just as with rolling a number cube.

• Students can create a simulation on a computer to extend the experimental results to 1,000 trials and compare the results to the theoretical probability.

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(8.12) Probability and statistics. The student uses statistical procedures to describe data.

(8.12.a) Probability and statistics. The student uses statistical procedures to describe data. The student is expected to select the appropriate measure of central tendency or range to describe a set of data and justify the choice for a particular situation.

Clarifying Activity with Assessment Connections

Students use a list of their grades to find the mean, median, mode, and range, and then select the measure of central tendency that describes their overall grade most favorably. Students are asked to present the data in such a way as to convince the teacher to give the student the most favorable grade.

Assessment Connections

Questioning . . .

Open with . . .

• Which measure of central tendency describes your grade most favorably?

Probe further with . . .

• How do the various measures differ? Which is the highest? Which is the lowest?
• Are there any grades that might skew the data?
• How do these outliers affect the data?
• Which measure describes your grade least favorably?
• What is it about the data that causes the measures to differ (e.g., why is the mode higher than the mean)?
• What measure do you think describes your grades most accurately?

Listen for . . .

• Is the student able to articulate his or her choice?
• Is the student using appropriate vocabulary when describing the data?
• Is the student correctly identifying the different measures?
• Is the student able to explain how each measure is related to the data?
• When selecting the measure to be used, does the student realize that the measure that shows the grades most favorably may not be a true reflection of his or her grades?

Look for . . .

• Is the student finding the measures in an organized and appropriate way?
• Does the student select a method for displaying the data to help find the measures?
• Is the student using graphs to represent the data?
• Does the student know what data might be causing a difference in the outcomes of the measures?
• Does the student recognize which of the measures would be least representative of the data if a report card grade were given?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 17
• Spring 2004, grade 8, item 18
• Spring 2006, grade 8, item 26

(8.12.b) Probability and statistics. The student uses statistical procedures to describe data. The student is expected to draw conclusions and make predictions by analyzing trends in scatterplots.

Clarifying Activity with Assessment Connections

Students time each other to see how many times they can jump a rope in one minute and in two minutes. They collect data, display the data in a table, and then use a scatterplot to graph the relationship between time and how many jumps a person can complete. Students then use the trend of the data on the scatterplot to predict how many jumps a person could do in 5 minutes, 10 minutes, etc.

Assessment Connections

Questioning . . .

Open with . . .

• Can you use a scatterplot to make predictions?

Probe further with . . .

• How many jumps did you complete in one minute? Two minutes?
• Can you predict how many you will be able to do in 10 minutes?
• What other factors affect the number of jumps you are able to complete?
• What are the trends in your data?
• What is the overall relationship between the number of jumps and time?

Listen for . . .

• Is the student able to identify patterns in the data?
• Can the student interpolate jumps in 2 1/2 minutes or 4 1/2 minutes?
• Is the student able to make generalizations based on the trends in the data?
• Is the student able to justify his or her predictions?

Look for . . .

• Is the student able to record the data and make a scatterplot?
• Does the student use technology to graph?
• Does the student see the trends in the scatterplot?
• Does the student draw a line of best fit?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 11
• Spring 2006, grade 8, item 4

(8.12.c) Probability and statistics. The student uses statistical procedures to describe data. The student is expected to select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, with and without the use of technology.

Clarifying Activity with Assessment Connections

Students collect and organize information about the amount of time they spend on homework (in each of their classes) for two weeks. Students brainstorm questions that can be answered with this data and make recommendations about what type of display would represent this data most effectively.

Students then decide on an appropriate representation for the data and justify their selection.

For example, "Circle graphs are good to show you what fraction of the homework is from each class and make it easy to compare the time spent on homework from each different class."

Assessment Connections

Questioning . . .

Open with . . .

• What are the things that you need to consider before constructing your display? What tools will you need?

Probe further with . . .

• What kinds of data do you have to represent?
• What kinds of representations might be helpful to describe the data? Why?
• What representations would not be appropriate? Why?
• Can you represent your data using fractions? Decimals? Percentages?
• How many degrees are there in a circle?
• How is your data represented in your circle graph?
• Can you use your circle graph to make predictions for a larger number of students?
• What information would you need to construct a bar graph or a histogram?

Listen for . . .

• Is the student able to describe and justify a decision to use a particular type of representation?
• Is the student able to articulate the relationship between the representation and the original data?
• Is the student able to make predictions for a larger group?
• Is the student considering the data and the questions as they make their selection?
• Is the student verbalizing a justification for the representations he or she chose?

Look for . . .

• Is the student able to recognize that the data can be represented in a variety of ways?
• Can the student construct line plots, line graphs, stem-and-leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams with and without the use of technology?
• Is the student using appropriate representations of data?
• If the student starts with an inappropriate representation, does he or she recognize the problem?
• Is the student's representation reasonable?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 4
• Spring 2006, grade 8, item 1

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(8.13) Probability and statistics. The student evaluates predictions and conclusions based on statistical data.

(8.13.a) Probability and statistics. The student evaluates predictions and conclusions based on statistical data. The student is expected to evaluate methods of sampling to determine validity of an inference made from a set of data.

Clarifying Activity with Assessment Connections

Ms. Trevino's class wants to find out how many students in their school have pets. The class divides into four groups and each group devises a plan for sampling the school population.

Group 1 will publish a notice in the school's weekly paper asking for volunteers for their survey. Each member of Group 2 will survey the students in his or her English class. Group 3 will survey every fifth person as they come in the building before school. Group 4 will randomly select 25 students from a list of 4-digit student ID numbers. They will roll a 10-sided number cube four times to generate each number.

Assessment Connections

Questioning . . .

Open with . . .

• Which sampling method will most accurately represent the number of students who have pets?

Probe further with . . .

• What are the advantages and disadvantages of each plan?
• How could each of these methods lead to invalid conclusions?
• Is it possible that one or more of these methods could lead to a biased sample?
• Are all these sampling methods fair? Are all students able to participate in each method?

Listen for . . .

• Is the student able to justify a choice of sampling method?
• Is the student able to analyze different sampling methods for fairness or bias?

Look for . . .

• Is the student able to select and justify an appropriate sampling method?
• Is the student able to recognize the flaws of other suggested sampling methods?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2006, grade 8, item 23

(8.13.b) Probability and statistics. The student evaluates predictions and conclusions based on statistical data. The student is expected to recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.

Clarifying Activity with Assessment Connections

Students use graphs from sources such as the local newspaper to evaluate the conclusions made and determine the validity of the information depicted on the graph. For example, if the units on the scale are not all the same size, then the visual representation of the data is distorted.

Assessment Connections

Questioning . . .

Open with . . .

• Are the conclusions accurate in each of the articles you read?

Probe further with . . .

• How did you decide if the information was accurate or inaccurate?
• What information is given in the graph?
• What relationship does the graph represent?
• Can you name what this ordered pair represents in the graph?
• What are the trends in the graphs?
• If the scale were adjusted, how would graph be different?
• Why would companies or organizations choose to represent their data with a certain type of graph?

Listen for . . .

• Is the student able to justify his or her conclusions?
• Is the student able to make predictions based on his or her conclusions?
• Is the student able to interpret the graph?

Look for . . .

• Is the student able to write statements that support his or her conclusions?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 16
• Spring 2004, grade 8, item 45

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(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

(8.14.a) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.2.D, Number, operation, and quantitative reasoning. Additional activities that exemplify this student expectation include 8.1D, 8.2D, 8.11B, 8.11C.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2006, grade 8, items 29, 34, and 48

(8.14.b) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.5B, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 8.2C, 8.3C, 8.9A, 8.11A, 8.11B, 8.13A.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 8, item 33
• Spring 2004, grade 8, item 12
• Spring 2006, grade 8, item 42

(8.14.c) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.8C, Measurement. Additional activities that exemplify this student expectation include 8.1A, 8.3B, 8.4, 8.6A, 8.10A, 8.11A.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, items 34 and 46
• Spring 2006, grade 8, item 25

(8.14.d) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.2B, Number, operation, and quantitative reasoning. Additional activities that exemplify this student expectation include 8.1B, 8.1C, 8.3B, 8.8A, 8.8B, 8.9B, 8.10B.

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(8.15) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models.

(8.15.a) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. The student is expected to communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.3A, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 8.2A, 8.2B, 8.7A, 8.8C, 8.12C, 8.13B.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, items 9 and 31
• Spring 2006, grade 8, item 6

(8.15.b) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. The student is expected to evaluate the effectiveness of different representations to communicate ideas.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.12C, Probability and statistics. Additional activities that exemplify this student expectation include 8.4, 8.11C, 8.12A, 8.12B, 8.12C, 8.13A.

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(8.16) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions.

(8.16.a) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to make conjectures from patterns or sets of examples and nonexamples.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.10B, Measurement. Additional activities that exemplify this student expectation include 8.5B, 8.6B, 8.7C, 8.8C, 8.10A.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 10
• Spring 2006, grade 8, items 9 and 39

(8.16.b) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to validate his/her conclusions using mathematical properties and relationships.

Clarifying Activity with Assessment Connections

See the Clarifying Activity for 8.8B, Measurement. Additional activities that exemplify this student expectation include 8.3B, 8.5A, 8.7B, 8.7C, 8.8C.

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 8, item 29
• Spring 2006, grade 8, items 8 and 12

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