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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.

(7.1) Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms.

(7.1.a) Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms. The student is expected to compare and order integers and positive rational numbers.

Clarifying Activity with Assessment Connections

The teacher sets up a table for each group of four or five students. Each table has a large, blank number line with no marks or numbers. Each student is assigned one non-negative integer or positive rational number (in fractional or decimal form). Each student has several notecards with that number written on them.

In each group, students compare their assigned numbers and place them on their number line. On a signal from the teacher, students rotate to the next table and add their assigned numbers to that group's number line. Students may need to rearrange the position of previously placed cards. Students continue the rotation until each group has placed their assigned numbers on each number line.

Assessment Connections

Questioning . . .

Open with . . .

• How do you determine where to place your number on the number line?

Probe further with . . .

• What strategies can you use to locate the position of your number?
• What is the greatest number? How do you know?
• What is the least number? How do you know?
• What benchmark is the most helpful in placing numbers on the number line? Explain.
• Do you need to rearrange previously placed numbers to appropriately place your number? Why or why not?
• What do you notice about the placement of your number in relation to other numbers?
• How did the group decide to place the numbers?

Listen for . . .

• Does the student verbalize patterns such as the number always being before, between, or after a particular number?
• What strategies does the student use when deciding where to place the number?
• Does the student compare the number in a variety of equivalent forms?

Look for . . .

• Does the student always attempt to place the assigned number at a specific place on each number line (e.g., 3 inches from the left, exactly in the middle, etc.)?
• Can the student position the number without assistance?
• Does the student check for reasonableness of number placement?

TAKS Connections

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

• Spring 2003, grade 7, item 26
• Spring 2004, grade 7, item 38
• Spring 2006, grade 7, item 29

(7.1.b) Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms. The student is expected to convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator.

Clarifying Activity with Assessment Connections

Students set up a table on their own paper that they will use to record their conversions; the table should have columns for Fraction, Decimal, and Percentage. Have various models available for students to use to assist them in their conversions such as number lines, decimal squares, fraction strips, etc.

Show students a number and ask them to convert that number to one of the other two forms on their own. Call on students to share their conversions and discuss the different ways in which the original number was represented. Look for interesting similarities and differences among student responses; for example, 4 and 3/8 could be represented 4.375 or 4.3750 or 437.5%, etc. Also encourage students to share and compare how they made their conversions by asking them questions such as, "What process or strategy did you use to convert this number?" and "How is your method of conversion similar to or different from the process used by other classmates?"

Allow students to check and compare their conversions using a calculator. Then discuss the ways in which the various representations of that number would be useful in the real world; for example, "When would it make more sense to use the fraction representation?" [Measurement of building materials, fabric, ingredients, stock market etc.]

Assessment Connections

Questioning . . .

Open with . . .

• What strategies can you use to convert numbers from one form to another form—fraction to decimal, decimal to fraction, fraction to percentage, percentage to decimal, etc.?

Probe further with . . .

• Is there more than one way to represent a fraction as a decimal? A decimal as a percentage? A percentage as a fraction? If so, provide examples.
• How can you use estimation to help you convert a number from one form to another form?
• How can you use a model (number line, decimal square, fraction strip, etc.) to convert between fractions, decimals, and percentages?
• How can you prove that two forms of a number are equivalent?
• How can you check to see if your strategy worked?
• Are there some numbers or types of numbers that are easier or harder to convert? Why?
• Are there some fractions that convert only to non-terminating decimals?

Listen for . . .

• Is the student able to articulate a strategy for making conversions?
• Is the student able to justify that two numbers are equivalent or approximately equivalent?
• Is the student able to use estimation to convert from one form of a number to another form?

Look for . . .

• Can the student extend his or her understanding of conversions to other fractions, decimals and percentages?
• Is the student able to use a model to justify that two numbers are equivalent or approximately equivalent?

TAKS Connections

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

• Spring 2004, grade 7, items 14 and 31
• Spring 2006, grade 7, item 18

(7.1.c) Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms. The student is expected to represent squares and square roots using geometric models.

Clarifying Activity with Assessment Connections

Students use tiles to construct square models and explore the relationship between the lengths of the sides and the areas of the models, and the square numbers and their square roots. For example,

Since 9 square tiles can form a square with the area of 9 square units, then 9 is a square number. The length of the side of the square, 3, is the square root of 9 ( 3 x 3 = 3² = 9).

Assessment Connections

Questioning . . .

Open with . . .

• What do you know about the length of the side in relation to the area of each square?

Probe further with . . .

• If you know that a square has an area of 9 and another square has an area of 16, what do you know about the length of a side of a square with an area between 9 and 16? Use a calculator to check your answer.
• What patterns can you see in the square models you built? In the lengths? In the areas?
• Does the side length of a square have to be a whole number? Why or why not?
• What are the attributes of a square? Which of these attributes will help determine the square root (length of the sides)?
• What is the definition of a square?

Listen for . . .

• Does the student recognize the unique relationship between length of sides and area in terms of square and square root?
• When the student finds the square root of a number that is not a perfect square does the student use benchmarks of perfect squares to determine the answer?

Look for . . .

• Is the student squaring numbers properly?
• Does the student have an organized method to determine square roots or squares?

TAKS Connections

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

• Spring 2003, grade 7, item 39
• Spring 2004, grade 7, item 33
• Spring 2006, grade 7, item 39

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(7.2) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

(7.2.a) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to represent multiplication and division situations involving fractions and decimals with models, including concrete objects, pictures, words, and numbers.

Clarifying Activity with Assessment Connections

Students are presented with the situation of a bake sale at school. Pans of cake are being sold. Present students with questions such as, "During a class bake sale Sam had 3/4 of a pan of chocolate cake left to sell at 2 p.m. An hour later he had sold 1/3 of that amount. How much of the whole cake was left after that hour?"

Students solve the problem situation by modeling with pictures and describe their solutions using words and numbers.

Example: Students draw a rectangular model to demonstrate 2/3 x 3/4 by first drawing a rectangle and shading 3/4.

Students then divide the rectangle into thirds in the opposite direction and shade 2/3.

Students describe the double-shaded portion as 2/3 of 3/4, or 2/3 x 3/4 = 6/12 = 1/2.

Assessment Connections

Questioning . . .

Open with . . .

• What strategies can you use to solve this problem?

Probe further with . . .

• Can you draw a picture to represent the cake Sam started with at 2 p.m.?
• Can you show the amount of the cake Sam sold in an hour?
• How can you use your picture to show the amount of the entire cake sold during the hour?
• Can you connect the picture to the number sentence?

Listen for . . .

• Can the student connect the model to the multiplication fact?
• Can the student describe the different shaded regions of the rectangle?
• Does the student recognize the double-shaded region as the product?
• Does the student recognize that both numerators and denominators were multiplied?
• Does the student recognize patterns in the models?
• Can the student justify his or her answers?

Look for . . .

• Can the student draw the correct representations for the situation?
• Does the student look for patterns in what is happening to the numerators and denominators of the fractional parts?
• Does the student recognize that when fractions of less than 1 are multiplied, the product is not bigger than the fractions?
• Can the student use mathematical symbols and operations to represent what he or she has modeled?
• Does the student recognize that the solutions in the model and the multiplication sentence with symbols are the same?

TAKS Connections

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

• Spring 2003, grade 7, item 10
• Spring 2004, grade 7, item 24
• Spring 2006, grade 7, item 10

(7.2.b) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals.

Clarifying Activity with Assessment Connections

Students imagine the following situation and work together to decide on the best course of action:

Two brothers, James and Diego, went shopping after school one day. Their mother said she would give each of them $60 to spend and, assuming that each item they purchased was on sale, she would pay the sales tax. Before going to the register, James and Diego had to calculate the total price of the purchases before tax to make sure they had not gone over the $60 limit. Below is a list of the original prices and the percentage or fractional discount of each item. Help James and Diego determine if they have selected items whose total cost is less than $60. Explain how you know.

James:

• Pants: $40, on sale for 1/2 off
• Shirt: $25, on sale for 30% off
• Shoes: $30, on sale for 15% off

Diego:

• Sweatshirt: $30, on sale for 1/4 off
• Jeans: $22, on sale for 20% off
• Shirt: $27, on sale for 1/3 off

Assessment Connections

Questioning . . .

Open with . . .

• How do you determine the sale price?

Probe further with . . .

• What information do you need to know to solve this problem?
• What operations could you use to solve this problem?
• How do you calculate the amount of the discount?
• How do you calculate the sale price?
• How do you calculate the total price of the three articles of clothing?
• Can you solve the problem in more than one way?

Listen for . . .

• Can the student support the reasonableness of his or her answer?
• What methods does the student use to compute the sale prices?
• Does the student find the sale price of each item?
• Does the student round appropriately?

Look for . . .

• Does the student use correct mathematics?
• What operations does the student use?
• Does the student use fractions or decimals consistently, or use them interchangeably when solving the problem?
• Does the student accurately determine the new price per item?
• Does the student use operations with decimals correctly?
• Does the student use operations with fractions correctly?

TAKS Connections

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

• Spring 2003, grade 7, item 1
• Spring 2006, grade 7, item 21

(7.2.c) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms.

Clarifying Activity with Assessment Connections

Students use models of negative and positive values (for example, counters of different colors, where red equals +1 and yellow equals -1) to demonstrate addition, subtraction, multiplication, and division of integers. Students collect their observations about the models and write a rule for each operation. For example, to show -3 - 2 = -5, students display 3 yellow counters for -3 and 2 zero pairs of red and yellow counters in order to be able to take away two red counters, leaving 5 yellow counters.

Students connect the results of this action to the algorithm for subtracting integers:

-3 - 2 = -2 + -3 = -5.

For multiplication and division:

Using a number line, students establish a pattern for multiplying numbers. For example, given the number sentences in the first column, students should notice a decrease of 3 for each step of the pattern:

Then students draw a number line and establish the pattern.

The teacher should remind students about the commutative property of multiplication before going to the next step. (-2 x 3 = 3 x -2)

Using the same kind of strategy, students try the activity again to model multiplication where both factors are negative. For example,

Again students draw a number line and establish the pattern.

Assessment Connections

Questioning . . .

Open with . . .

• What situations result in answers that are positive? What situations result in answers that are negative?

Probe further with . . .

• When you add two integers, can you tell if the sum is positive or negative? How?
• When you subtract two integers, can you tell if the difference is positive or negative? How?
• When you multiply integers, can you tell if the product is positive or negative? How?
• When you divide integers, can you tell if the quotient is positive or negative? How?
• How is adding integers similar to multiplying integers? How is it different?
• How is subtracting integers similar to dividing integers? How is it different?
• What do you notice when you model the situations?

Listen for . . .

• Can the student verbalize rules regarding integers for each of the operations?
• Does the student recognize patterns when modeling?
• Can the student decide if the answer is positive or negative without calculating the problem?
• What strategy does the student use to find the solution?
• Can the student verbalize the algorithm?

Look for . . .

• Does the student make conclusions or generalizations about the operations using integers?
• Does the student recognize patterns that help him or her predict if the solution is positive or negative?
• Can the student move from the model to a generalization?
• Does the student consider the reasonableness of the solution?
• Can the student demonstrate and explain the solutions?

TAKS Connections

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

• Spring 2004, grade 7, item 27
• Spring 2006, grade 7, item 38

(7.2.d) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes, and student-teacher ratio.

Clarifying Activity with Assessment Connections

Students imagine the following situation and work together to decide on the best course of action:

The student council is planning a party. The group decides to use a grocery store advertisement to help them with their choices. Carl decides he will buy the food if everyone agrees to share in the cost. Because some items have been donated, the group needs to buy only sodas and hotdogs.

Given the advertisement information, help Carl decide which items to buy in order to save the most money. Write Carl a note explaining how to compare the prices and what are the best buys. As a school organization, the student council will get a 7 1/2% discount from the store and will not be charged tax. Help Carl determine how much money to collect from each of the 25 students. Explain your reasoning.

Soda:

• Brand A: 12-pack $2.75
• Brand B: 6-pack $1.29
• Brand C: 24-pack $5.60

Hotdogs:

• One package of 10 for $2.19
• Buy one package of 10 for $2.79 get the second package for 1/3 off

Assessment Connections

Questioning . . .

Open with . . .

• How do you determine what to buy? Why?

Probe further with . . .

• What information do you need to know to solve this problem?
• What operations could you use to solve this problem?
• How do you compare prices?
• Does the number of students you buy for affect your choice?
• What type of rational numbers will you be working with?
• Can you solve the problem in more than one way?

Listen for . . .

• Can the student support the reasonableness of his or her answer?
• What methods does the student use to compare prices?
• Does the student discount the amount by 7 1/2%?
• Does the student round appropriately?

Look for . . .

• Does the student use correct mathematics?
• What operations does the student use?
• Does the student use fractions or decimals consistently, or use them interchangeably when solving the problem?
• Does the student accurately determine the price per item?
• Does the student use operations with decimals correctly?
• Does the student use operations with fractions correctly?

TAKS Connections

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

• Spring 2003, grade 7, item 34
• Spring 2004, grade 7, item 1
• Spring 2006, grade 7, item 3

(7.2.e) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to simplify numerical expressions involving order of operations and exponents.

Clarifying Activity with Assessment Connections

The teacher writes the expression (3 + 2)² ÷ 6 - 2 + 4 on the board. Each student creates cards with each symbol and number in the expression. The teacher asks the students to rearrange the cards to create a new expression according to one of the guidelines below.

• Change the numbers so you have a greater value.
• Change the grouping symbols so you have a smaller value.
• Change the operations so you have a greater value.
• Change both grouping and operations to have a smaller value.
• Using every card, what is the expression that will yield the largest value? smallest value?

Assessment Connections

Questioning . . .

Open with . . .

• What is the value of the given expression?

Probe further with . . .

• What do you do first to simplify the expression? Next?
• What happens if you change the order of the numbers or symbols?
• How could you change the value of the expression to make the value greater? Smaller?

Listen for . . .

• Can the student verbalize the correct use of parentheses?
• Can the student summarize findings concerning order of operations?

Look for . . .

• Does the student construct expressions according to the guidelines given?
• Can the students use parentheses appropriately?
• Does the student self-monitor and self-correct?

TAKS Connections

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

• Spring 2003, grade 7, item 30
• Spring 2004, grade 7, item 30
• Spring 2006, grade 7, item 44

(7.2.f) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to select and use appropriate operations to solve problems and justify the selections.

Clarifying Activity with Assessment Connections

The teacher places student-created problem situations on an overhead projector. Students are asked to select the appropriate operations to solve the problems, and then to justify their reasoning. Students then compare justifications.

Assessment Connections

Questioning . . .

Open with . . .

• What strategy would you use to solve the problems?

Probe further with . . .

• What operation do you need to perform?
• Would a different operation work?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Can the student verbalize the connections between the problem and the operation required?

Look for . . .

• Is the student using the correct operation?
• Can the student perform the operation correctly?

TAKS Connections

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

• Spring 2003, grade 7, item 37
• Spring 2006, grade 7, item 12

(7.2.g) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to determine the reasonableness of a solution to a problem.

Clarifying Activity with Assessment Connections

The teacher gives students the following scenario:

Your grandmother from out of state wants to buy you a birthday outfit. She wants you to tell her how much money you need to buy the outfit. Using advertisement flyers from department stores or online catalogs, select an outfit. Using estimations, determine a reasonable amount of money you will need to make the purchases, then write a short note to your grandmother explaining the amount of money you will need.

Assessment Connections

Questioning . . .

Open with . . .

• What is your plan for determining the amount of money you need?

Probe further with . . .

• Do you need an exact answer for this situation?
• Is your request reasonable?
• What makes your request reasonable?
• When would you need more money?
• When would you need less money?
• What would you do to make sure you have enough money to cover the cost of the outfit?

Listen for . . .

• Is the student estimating prices?
• What are the operations the student is using to determine the amount of money he or she needs?
• If the student buys multiple items does the student multiply or add?
• Does the student ask for a reasonable amount?
• Does the student write a note explaining the amount of money needed?
• Can the student justify his or her answer?

Look for . . .

• Is the student using prices based on the advertisement or catalog?
• When does the student round dollar amounts?
• Does the student follow a procedure when determining the amount of money needed?
• Is the student computing the amount needed correctly?

TAKS Connections

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

• Spring 2003, grade 7, item 48
• Spring 2004, grade 7, item 45
• Spring 2006, grade 7, item 23

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(7.3) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships.

(7.3.a) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to estimate and find solutions to application problems involving percent.

Clarifying Activity with Assessment Connections

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

Mary and Cindy are at the store shopping for scooters. The deluxe model costs $139.95, and the basic model costs $79.95. The girls plan on combining their money to buy two scooters. Mary has $120.00 and Cindy has $70.00. The store is offering different discounts for early shoppers.

• Between 8 a.m. and 9 a.m., 20% off
• Between 9 a.m. and 10 a.m., 15% off
• Between 10 a.m. and 12 p.m., 10% off

Using estimation, determine what combination of scooters could be purchased with and without the discount. Based on the amount of money Mary and Cindy have, which scooters can they purchase, and at what time would they save the most amount of money? Create a table of your information to support your choices.

Example of a table students could create:

Click here for a larger version of this table.

Assessment Connections

Questioning . . .

Open with . . .

• What combination of scooters can the girls afford to buy? What information did you use to make your decision?

Probe further with . . .

• What amounts will you use for your estimates?
• How do you calculate the amount of discount?
• How do you calculate the sales price?
• How do you calculate the total cost of each scooter?
• How do you find the percentage of a number?
• How do your estimates compare to the actual costs?
• Can you buy two of the deluxe models at any time?
• Can you buy one of each kind of scooter at any time?
• When would you get the best deal?

Listen for . . .

• Does the student verbalize the operations used?
• Does the student verbalize the justification?
• Does the student use mental math?
• Does the student check for reasonableness?
• Does the student recognize the patterns in the table?

Look for . . .

• Is the student using benchmarks (with percentages of 10%, 15%, 20%)?
• Does the student record necessary information?
• Does the student use a table to record the information appropriately?
• Does the student use appropriate operations?
• Does the student use calculators when appropriate?

TAKS Connections

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

• Spring 2003, grade 7, item 17
• Spring 2004, grade 7, item 25
• Spring 2006, grade 7, item 34

(7.3.b) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit costs, and related measurement units.

Clarifying Activity with Assessment Connections

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

The school is having its annual Mexican dinner fundraiser. This year we expect 200 people to attend the event. Your club will provide the salsa. You will make salsa with three levels of spiciness: hot, medium, and mild. The recipes are listed below. Each recipe will make 10 servings. How will you decide the number of peppers and tomatoes you will need to make enough salsa for the 200 people?

Click here for a larger version of this table.

Assessment Connections

Questioning . . .

Open with . . .

• Using the information given, how will you decide the number of peppers and tomatoes you will need?

Probe further with . . .

• How many people do you think are going to want hot salsa? Medium salsa? Mild salsa?
• How many batches of each type of salsa will you need?
• What ratios will you use to determine the number of batches you will need?
• How many different types of recipes will be made?
• What are the ratios you will use for each recipe?
• How many times will you need to make each recipe in order to serve 20 people? 30 people?
• What is meant by a proportional relationship?
• What proportional relationship will help you determine the number of recipes you need to serve all the people?
• What proportional relationship can you use to solve this problem?
• How can you verify that your proportional relationship is set up correctly?
• A survey determined that 1/2 the people like hot salsa. If you make 1/2 of the salsa hot, how many peppers and tomatoes will you need to have the right amounts of salsa for the 200 people?
• What are the percentages of tomatoes and peppers in each recipe?

Listen for . . .

• Can the student use correct ratios?
• Can the student set up a proportional relationship?
• Can the student describe the proportional relationship that was set up?
• Can the student identify the units in the proportional relationship?
• What strategies does the student use to set up a proportional relationship?
• Can the student describe the process used to solve the problem?
• Can the student justify the solution?
• Can the student find the percentages of tomatoes and peppers in each recipe?

Look for . . .

• Can the student label the quantities in the proportional relationship?
• Can the student represent the proportional relationship symbolically?
• Can the student use an organizational device to record the data?
• Can the student use a systematic process to answer the question?
• Can the student reason through the proportional relationship?
• Does the student check for reasonableness in the amounts of peppers and tomatoes?
• Can the student find the percentages of tomatoes and peppers in each recipe?

TAKS Connections

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

• Spring 2003, grade 7, item 28
• Spring 2004, grade 7, item 47
• Spring 2006, grade 7, item 43

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(7.4) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form.

(7.4.a) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. The student is expected to generate formulas involving unit conversions, perimeter, area, circumference, volume, and scaling.

Clarifying Activity with Assessment Connections

Each student forms a rectangle from a 20-centimeter length of wire, ribbon, pipe cleaners, or similar item. The ends must meet at a corner of the rectangle.

In small groups, students share their strategies for building their rectangles, compare their rectangles, and make sure that they have a variety (short and wide, skinny and long, square). On each rectangle students mark the corners with a pen or marker. Each student then straightens his or her wire or material they used and the group compares all the marks.

Each group creates a table to record each rectangle's dimensions, perimeter, and area. Students then determine rules for all of the patterns they observe in their tables.

Assessment Connections

Questioning . . .

Open with . . .

• What generalizations can you make from the patterns in the table?

Probe further with . . .

• What information should be included in the table?
• What do you notice about the marks on the wires?
• What do you notice about the lengths of the wires?
• Do all of the wires have the same markings?
• What do the marks represent?
• What patterns do you see in the table?
• For each length and width, find the sum. What do you notice?
• If you know one side is 2.8 centimeters, what can you conclude about the other sides' measurement?
• Can you write a sentence that describes the relationship between the numbers in your table and the length of the wire?
• Can you write a mathematical rule (symbolically) that describes the relationship between the numbers in your table and the length of the wire (e.g., l + w = (1/2) x 20, l + w = 10, 10 - l = w, 10 - w = l )?
• What if your wire was 30 centimeters in length? How would that change your rule?

Listen for . . .

• Is the student using appropriate units of measure?
• Is the student using appropriate vocabulary?
• Does the student recognize the relationship between the length of sides and the perimeter?
• Does the student recognize the relationship between length of sides and the area?
• Can the student communicate about patterns in the table?
• Can the student communicate the generalization?

Look for . . .

• Do all wires have the midpoint marked correctly?
• Is the student measuring accurately with a ruler?
• Is the student using appropriate units of measure?
• How is the student organizing his or her information in a table?
• Is the student generalizing the patterns?
• Is the student using variables in the generalization?

TAKS Connections

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

• Spring 2003, grade 7, item 15
• Spring 2006, grade 7, items 30 and 41

(7.4.b) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. The student is expected to graph data to demonstrate relationships in familiar concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

Clarifying Activity with Assessment Connections

Using the results of the activity in 7.4A, students graph data for length and width of the rectangles formed from 20-centimeter length of wire (or other material).

Assessment Connections

Questioning . . .

Open with . . .

• What relationships are represented in your graph?

Probe further with . . .

• How are you labeling your axes?
• How are you scaling your axes?
• What does an ordered pair represent on your graph?
• What is the symbolic rule for your graph?
• What does your rule mean?
• Describe your graph (e.g., its appearance, shape, direction, etc.).
• How can a graphing calculator be used to display this information?
• If you know one dimension of a rectangle, how can you use the graph to get the other dimension?
• How can you determine the perimeter of the rectangle from the graph?
• Is 20 centimeters a reasonable dimension for the rectangle? Why or why not?
• As one dimension increases, what do you notice about the other dimension?
• When the dimensions on the graph are the same, what kind of special shape do you have?
• If the perimeter of the rectangle is 30 centimeters, how does that change your graph?
• How are the table, graph, and symbolic rule related?

Listen for . . .

• Is the student using appropriate vocabulary to describe the dimensions?
• Does the student use words like table, ordered pair, axes, graph, plot, dependent and independent variable, and so on when discussing representations?
• How does the student verbally describe the numerical relationship between the lengths of the sides of the rectangle? (For example, "As one increases the other decreases.")
• Can the student verbally describe the connections between the table, the graph, and the rule? (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 organized method to record the dimensions of the rectangle?
• Does the student self-correct errors?
• Does the student check for reasonable units of measure (length) on the graph?
• Does the student accurately represent the information?
• Does the student label the axes correctly?
• Does the student use the graphing calculator to represent the information?
• Can the student use the graphing calculator to create lists and scatter plots?

TAKS Connections

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

• Spring 2003, grade 7, item 36
• Spring 2004, grade 7, item 19
• Spring 2006, grade 7, item 31

(7.4.c) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. The student is expected to use words and symbols to describe the relationship between the terms in an arithmetic sequence (with a constant rate of change) and their positions in the sequence.

Clarifying Activity with Assessment Connections

The teacher gives students the following scenario:

Joe spends time each day building brick towers. Each day he builds one more layer on his tower.

If Joe continues this pattern, write a rule that will help him find the number of bricks that will be in the tower on the 100th day.

Assessment Connections

Questioning . . .

Open with . . .

• Using the figures given, how will you determine the number of bricks in the tower on the fourth day? Fifth day? Tenth day? 100th day?

Probe further with . . .

• What patterns do you see?
• Do you need to draw every figure from day 4 to day 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?
• What is the relationship between a given day and the number of bricks in the tower?
• How can you use this relationship to help you find the number of bricks that will be in the tower on the fourth and fifth days?
• What generalizations can you make?
• If the tower has 33 bricks, what day is it?
• If the tower has 219 bricks, what day is it?
• Can you use a graphing calculator to test your rule? How?
• How can you organize the information?

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?
• Does the student differentiate the term number from the number of bricks?

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?
• 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 7, item 20
• Spring 2006, grade 7, items 5 and 37

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(7.5) Patterns, relationships, and algebraic thinking. The student uses equations to solve problems.

(7.5.a) Patterns, relationships, and algebraic thinking. The student uses equations to solve problems. The student is expected to use concrete and pictorial models to solve equations and use symbols to record the actions.

Clarifying Activity with Assessment Connections

The teacher gives the students the following scenario to solve using a concrete or pictorial model:

Cindy's little brother removed all of Cindy's pencils from their boxes. Cindy came home to find 3 empty boxes and 18 pencils lying on the floor. She refilled all the boxes and had 3 pencils left over. How many pencils are in each box?

Cindy went to her brother's room and found all of his crayons out of their boxes. She found 6 empty boxes and 75 crayons on the floor. Cindy helped her brother refill all the boxes and there were 3 crayons left over. How many crayons are in each box?

Model these situations using algebra tiles. Write an equation from the model, and solve the equation.

Assessment Connections

Questioning . . .

Open with . . .

• How did you use the algebra tiles to solve the equation?

Probe further with . . .

• Which algebra tile matches your boxes?
• Which algebra tile represents your pencils or crayons?
• How can you write an equation using the model?
• Can you draw a picture of your model?
• What will you do with the leftovers?
• How are you grouping the pencils, crayons, and boxes?
• How many pencils and crayons are in each box?

Listen for . . .

• Does the student use appropriate vocabulary to describe the situation?
• How does the student describe verbally the numerical relationship between the total number of pencils and the number of pencils in a box?
• Can the student verbally describe the connections between the concrete model, the picture, and the symbolic representation?

Look for . . .

• Does the student organize information?
• Can the student represent the information with a model?
• Does the student use symbols to describe the model or picture?
• Does the student recognize flaws or errors in the model/picture/symbol representations and adjust them?

TAKS Connections

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

• Spring 2003, grade 7, item 31
• Spring 2006, grade 7, item 11

(7.5.b) Patterns, relationships, and algebraic thinking. The student uses equations to solve problems. The student is expected to formulate problem situations when given a simple equation and formulate an equation when given a problem situation.

Clarifying Activity with Assessment Connections

Students play a game similar to "Jeopardy," with equations and problem situations under categories such as Sports. Students pick a category and value (such as Sports for 100) to reveal an equation or problem situation under the category. Given an equation, the students create a problem situation with the context described by the category. Given a problem situation, the students formulate a corresponding equation.

Examples:

In the Sports category, a student might get the equation 29 = 1 + 2x. The student might answer with the question "If you need two teams (with the same number of members) and one official to play a game, and there are 29 total people, how many people will be on each team?"

Or, the student might get the problem situation "Mark scored 29 points in last night's basketball game. He made one freethrow and the rest of his points were two-point baskets. How many two-point baskets did he make?" In this case, the student might answer, "What is x if 29 = 1 + 2x?"

Assessment Connections

Questioning . . .

Open with . . .

• How did you devise your question?

Probe further with . . .

• In your situation, what does the 29 represent?
• What does the 1 represent?
• What does the x represent?
• What does the 2x represent?
• Why do we add the 1 and the 2x?
• What is another way to represent the situation?

Listen for . . .

• Does student's language indicate that the student is relating the context to the symbols?

Look for . . .

• Can the student formulate problem situations when given a simple equation?
• Can the student formulate an equation when given a problem situation?

TAKS Connections

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

• Spring 2003, grade 7, item 16
• Spring 2004, grade 7, item 37
• Spring 2006, grade 7, item 7

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(7.6) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties.

(7.6.a) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to use angle measurements to classify pairs of angles as complementary or supplementary.

Clarifying Activity with Assessment Connections

Students play a "Find Your Partner" game in which each student has a picture of an angle and finds the student who has the angle supplementary (or complementary) to it. Students use angle measures to verify the choice. If geometry exploration software is available, students draw the adjacent angles to check for accuracy.

Assessment Connections

Questioning . . .

Open with . . .

• How can you tell if two angles are complements or supplements of each other?

Probe further with . . .

• What is an angle?
• What are the components of an angle?
• What are some ways to notate angles?
• What are adjacent angles?
• Do complementary or supplementary angles have to be adjacent?
• Do complementary or supplementary angles have to share a common ray?
• What are some visual clues for recognizing complementary or supplementary angles?
• How can you use angle measurements to prove if angles are complementary or supplementary?

Listen for . . .

• What language does the student use to prove pairs of angles are supplementary or complementary?
• Can the student name visual clues?
• Can the student name the ray shared by adjacent angles making supplementary or complementary pairs?
• Can the student name the vertex shared by adjacent angles making supplementary or complementary pairs?
• Is the student able to give the definition for complementary angles? Supplementary angles?

Look for . . .

• Is the student able to identify complementary and supplementary angles?
• Can the student match two angles to form complementary pairs? Supplementary pairs?

TAKS Connections

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

• Spring 2003, grade 7, item 13
• Spring 2004, grade 7, item 5
• Spring 2006, grade 7, item 42

(7.6.b) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to use properties to classify triangles and quadrilaterals.

Clarifying Activity with Assessment Connections

Students in cooperative groups use geometric vocabulary and properties to classify shapes. Each group will be given an assortment of shapes including triangles (scalene, equilateral, right, etc.) and quadrilaterals (squares, trapezoids, etc.). The groups create their own categories to classify shapes using geometric properties. Categories are arranged on a Venn diagram to be shared with the whole class. (Venn diagrams may be illustrated using geometric software.)

Assessment Connections

Questioning . . .

Open with . . .

• Explain how you formed the categories to classify the shapes.

Probe further with . . .

• How many categories did you use?
• What geometric terms did your group use?
• What helped you sort the shapes accurately?
• Did you find shapes that did not fit a category? Where did you place these shapes?
• Did any shapes have properties allowing them to belong to more than one category? How is this situation modeled in the Venn diagram?
• In a Venn diagram, is there a mathematical term that means an item belongs to more than one category?
• Could you narrow the shapes into three categories? Two categories?
• Is there a limit to the number of categories you can create?
• How did the Venn diagrams compare to each other?
• Have you learned some new properties from the other Venn diagrams?
• Would you organize your Venn diagram differently? Why or why not?

Listen for . . .

• Does the student verbalize a variety of geometric terms?
• Does the student analyze each shape with accurate vocabulary?
• Does the student formulate questions to explore geometric properties?
• Does the student use properties to synthesize group ideas into geometric categories?
• What strategies does the student use to determine the categories?

Look for . . .

• Does the student organize shapes for classification?
• Can the student construct a Venn diagram?
• Does the student label categories correctly?
• Does the student interact with group members to contribute vocabulary and geometric properties?

TAKS Connections

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

• Spring 2003, grade 7, item 40
• Spring 2004, grade 7, item 15
• Spring 2006, grade 7, item 14

(7.6.c) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders.

Clarifying Activity with Assessment Connections

In a cooperative group, students classify solids on a matrix of geometric properties. Each group will analyze a set of solids common to the student's environment (such as cereal boxes, salt boxes, soda cans, ice cream cones).

Through discussion and manipulation, students explore the structure of the solids to find their geometric properties. They then create a diagram to illustrate the matching of solids with their corresponding properties. To extend, each group may share their diagram with the class to discuss other properties or other characteristics of the solids.

Assessment Connections

Questioning . . .

Open with . . .

• Explain the classification system you used to organize your solids.

Probe further with . . .

• How does your previous knowledge of two-dimensional shapes relate to this activity?
• How do the properties of the solids help you understand the similarities and differences among the three-dimensional figures?
• How did knowledge of different types of angles help you with classifications?
• How do definitions of polygons relate to properties of solids?
• Can you use geometric vocabulary (such as congruent, parallel, dimensions, etc.) to assist with your description of the properties of the solids?
• What labels for categories of geometric properties did you use?
• What geometric properties are common to most solids? Are there geometric properties common to all solids?
• How do properties of a solid relate to how the object is used?

Listen for . . .

• Does the student verbalize a variety of geometric terms appropriate to three-dimensional figures?
• Does the student analyze each solid with accurate vocabulary?
• Does the student formulate questions to explore geometric properties?
• Does the student extend a mathematics idea from one solid to another?
• Does the student use properties to synthesize group ideas into geometric categories?
• What strategies does the student use to determine the categories?
• How does the student transfer characteristics of two-dimensional figures to the corresponding three-dimensional figures? (For example, circles are specific to cylinders, cones, and spheres.)

Look for . . .

• Can the student classify the solids into categories?
• Does the student justify conjectures?
• Can the student go from specifics to a generalization?
• Does the student demonstrate strategies in categorizing solids (such as using visual clues or measurement skills, or rotating the object)?

TAKS Connections

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

• Spring 2003, grade 7, item 29

(7.6.d) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to use critical attributes to define similarity.

Clarifying Activity with Assessment Connections

Students use geometric exploration software to stretch or shrink a simple shape so that it stays the same shape, but is a different size. Students use rulers and protractors (or measuring tools in the software) to compare the measurements of the corresponding angles and corresponding sides of the similar shapes. Students should notice that the pairs of corresponding angles are congruent and that the pairs of corresponding sides have a constant ratio. If software is not available, students should be given similar shapes so that they can compare measurements.

Assessment Connections

Questioning . . .

Open with . . .

• Tell me what makes two shapes similar.

Probe further with . . .

• How do the corresponding angles of the figures compare?
• How do the corresponding sides of the figures compare?
• What happens to the attributes of the figure as it gets smaller or larger?
• How do you express the relationship between two similar figures in a mathematical way?
• How do the perimeters compare?
• How do the areas compare?
• How do the ratios of the sides compare?

Listen for . . .

• Can the student articulate the changes in the figures?
• Can the student find patterns?
• Can the student describe the similarities?
• Can the student define similarity using mathematical terms such as scale factor, ratio, corresponding sides, etc.?

Look for . . .

• Does the student use measurement tools correctly?
• Does the student match two angles or two sides to make a corresponding pair?
• Can the student stretch or shrink the figure and retain the same shape?
• Does the student use measurements to find the attributes of similar figures?

TAKS Connections

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

• Spring 2003, grade 7, item 22
• Spring 2004, grade 7, item 9
• Spring 2006, grade 7, item 9

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(7.7) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane.

(7.7.a) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane. The student is expected to locate and name points on a coordinate plane using ordered pairs of integers.

Clarifying Activity with Assessment Connections

Students play a game of hide-and-go-seek on a coordinate plane. In pairs, students are given a coordinate plane. Each student identifies and marks three horizontally, vertically, or diagonally adjacent points. Students then take turns calling out ordered pairs, trying to locate the hidden points.

For example, if student A calls out an ordered pair in a quadrant different from where the points are hidden, student B must reply "big miss." If the ordered pair is within the same quadrant, but a miss, then the student B replies "miss." If the ordered pair locates one of the hidden points, student B replies "hit."

The object of the game is to locate the hidden points in as few tries as possible.

Assessment Connections

Questioning . . .

Open with . . .

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

Probe further with . . .

• What is a coordinate plane?
• What symbols and terms are needed to construct a coordinate plane?
• What is an ordered pair?
• What special notation is used to identify an ordered pair?
• Is the order of the numbers of an ordered pair important? Why?
• What determines the location of the ordered pair on the coordinate plane?
• What is unique about each quadrant in a coordinate plane?

Listen for . . .

• Does the student describe the coordinate plane with geometric terms such as x-axis, y-axis, point of origin, etc.?
• Does the student verbalize directional vocabulary (such as up, down, north, south, left, right, etc.) as represented on the coordinate plane?
• Can the student verbalize instructions to plot the ordered pair on the coordinate plane?
• Does the student identify the different quadrants of the coordinate plane?

Look for . . .

• Does the student match directions with integer value?
• Does the student plot the ordered pairs correctly?
• Can the student locate and name points on the coordinate plane?
• Can the student self-check for accuracy of the plotted points?

TAKS Connections

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

• Spring 2003, grade 7, item 44
• Spring 2004, grade 7, item 2
• Spring 2006, grade 7, item 1

(7.7.b) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane. The student is expected to graph reflections across the horizontal or vertical axis and graph translations on a coordinate plane.

Clarifying Activity with Assessment Connections

Given a simple drawing graphed on a coordinate plane, students are to move the drawing to another position on the coordinate plane without changing its shape. Students make a list of points they think will translate their drawing, plot the points, and test their conjecture. Or, with geometry exploration software, students can drag the drawing and identify its new set of points.

After students identify a correct set of points for the translation, they look for patterns between the pairs of coordinates that describe the translation. (A translation can be described by a constant change in the x coordinates, the y coordinates, or both. For example, adding 6 to the x coordinates and adding 1/2 to the y coordinates translates a figure to the right 6 units and down 1/2 of a unit.)

Students recreate the activity to reflect the simple drawing across the horizontal axis and then across the vertical axis. When the students look for patterns, they notice that when the drawing is reflected across the horizontal axis, the x-values remain the same and the y-values are the opposites of the originals. They also notice that when the drawing is reflected across the vertical axis, the y-values remain the same and the x-values are the opposites of the originals.

Assessment Connections

Questioning . . .

Open with . . .

• What did you notice as you graphed reflections across the horizontal or vertical axis and graphed translations on a coordinate plane?

Probe further with . . .

• What happens to the figure's size and shape when it is translated? Reflected? How are the size and shape different?
• What happens to the value of the coordinate pairs as the figure translates? Reflects across the x-axis? Reflects across the y-axis?
• What patterns do you notice?
• How do the values of the coordinate pairs in the original figure compare to coordinate pairs in the translation? In the reflections?
• How can you create the point values for a translation (or reflections) without drawing the figure?
• What conjectures can you create about translations (or reflections)?

Listen for . . .

• Does the student notice that the drawing of the translation has the same size and shape as the original figure?
• Does the student identify patterns that describe the translation and reflections?
• Can the student support conjectures made about the translation and reflections?
• Can the student describe constant changes in the x and y values?
• Does the student use the terms "translation" and "reflection" accurately or correctly?

Look for . . .

• Can the student graph reflections across the horizontal or vertical axis and graph translations on a coordinate plane?
• Can the student list the correct values for points of the translation and reflections?

TAKS Connections

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

• Spring 2004, grade 7, item 43
• Spring 2006, grade 7, item 13

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

(7.8.a) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to sketch three-dimensional figures when given the top, side, and front views.

Clarifying Activity with Assessment Connections

Given the top, front and side views of a solid, students use cubes to build all possible models that fit these views, then sketch the models on isometric dot paper.

Example: Groups of students build three-dimensional models with blocks, and then sketch the top, front, and side views. Groups switch 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 7, item 5
• Spring 2004, grade 7, item 39
• Spring 2006, grade 7, item 4

(7.8.b) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to make a net (two-dimensional model) of the surface area of a three-dimensional figure.

Clarifying Activity with Assessment Connections

Given a rectangular prism (a box), students use centimeter grid paper to make all possible nets to form the box. Students test each paper net by cutting it out and wrapping it around the box. Students then write descriptions for their nets.

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

Assessment Connections

Questioning . . .

Open with . . .

• How can you determine the net for the solid?

Probe further with . . .

• What are the geometric shapes that form the net?
• What strategies could you use to create your net?
• What shapes will you need to make the net? How many of each shape?
• Are there different ways to connect the shapes in the net so that it creates the same solid when folded?
• Can you find patterns among the nets?

Listen for . . .

• Can the student use geometric vocabulary to describe the shapes within the net?
• Can the student tell how many of each shape make up the net?
• Can the student tell how many faces make up the net?
• Can the student verbalize the strategies used?

Look for . . .

• Can the student create all the different nets for a solid?
• Does the student include all faces of the solid?
• Can the student fold the net to verify that it creates the desired solid?
• Does the student check for overlapping faces when the net is folded?
• Is the student using tools to carry out his or her strategy (rulers, grid paper, etc.)?

TAKS Connections

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

• Spring 2003, grade 7, item 46
• Spring 2006, grade 7, item 36

(7.8.c) 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 Activities:

Students use shape sets and the angle measures of polygons to determine whether a polygon can tile the plane (tessellate). Students then identify examples of tessellations.

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

Assessment Connections

Questioning . . .

Open with . . .

• How can you tell if a shape can tessellate, or tile a floor leaving no empty spaces?

Probe further with . . .

• Can all regular polygons be used to tessellate?
• What determines if a regular polygon tessellates?
• What are the angle attributes of a regular polygon?
• Can combinations of regular polygons tessellate?
• What are the angle measurements at the point where the corners of the polygons meet?
• How are tessellations used in art and architecture?

Listen for . . .

• Does the student verbalize an understanding of tessellations?
• Can the student identify regular polygons by name?
• Does the student analyze tessellations for patterns?
• Does the student form questions for trouble-shooting during the tessellation process?
• Does the student notice connections between tessellation and art and architecture?

Look for . . .

• Can the student measure angles correctly?
• Does the student self-check for accuracy of measurements?
• Does the student demonstrate knowledge of tessellations?
• Does the student record the results for the shapes that do and do not tessellate?
• Does the student document patterns of tessellations?
• Does the student combine a variety of shapes to tessellate?
• Does the student notate angle patterns found at the point where the corners of the polygons meet?

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(7.9) Measurement. The student solves application problems involving estimation and measurement.

(7.9.a) Measurement. The student solves application problems involving estimation and measurement. The student is expected to estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes.

Clarifying Activity with Assessment Connections

Students design possible efficient yet attention-grabbing candy boxes to hold a single layer of 24 candies, each in the shape of right triangles with 1-inch legs. They record the shape, dimensions, area of the base, and perimeter of the boxes.

Assessment Connections

Questioning . . .

Open with . . .

• What criteria did you use to determine efficiency of your box?

Probe further with . . .

• How does the efficiency of one box compare to the efficiency of another?
• What are the approximate dimensions of your box?
• Do you believe this will be an efficient box?
• How does the base of the box compare to the area of the candies that fill that base?
• Extension question: About how much air space is in your box along with the candies?

Listen for . . .

• Is the student using geometric language to discuss efficiency of the box?
• Does the student make conjectures about the size of the box?
• How reasonable are the conjectures? If not reasonable, does the student make modifications before proceeding?
• Is the student using mathematical language to discuss arrangements of cookies in the box?

Look for . . .

• Is the student's estimate of box size reasonable to the design chosen as efficient?
• Is the student recording the data for the possible boxes in an organized way?
• Is the student basing his or her justification on the efficiency criteria established?
• Is the student considering different dimensions of boxes?
• Is the student using appropriate units, tools, or formulas for finding areas and volume?

TAKS Connections

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

• Spring 2006, grade 7, items 8, 16, and 25

(7.9.b) Measurement. The student solves application problems involving estimation and measurement. The student is expected to connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders.

Clarifying Activity with Assessment Connections

Students design rectangular prisms that hold exactly 24 cubes that are 1 cm³. Using scissors, tape, index cards, and 1 cm³ cubes, they test their designs. They take note of the dimensions of the prisms and relate the measurements to the formulas for the volumes of the prisms.

Assessment Connections

Questioning . . .

Open with . . .

• What do you notice about the relationship between the dimensions of the rectangular prisms and their volume?

Probe further with . . .

• What are appropriate dimensions for your rectangular prism?
• What is the area of the base of the rectangular prism? How do you know?
• How many cubes will it take to fill one layer of your prism?
• How many layers will it take to fill your prism?
• If the base of your rectangular prism is 12 cm², what is its height? How do you know?
• If you want to build a prism that is twice as high but maintain the volume, how will this affect the area of the base?
• Can rectangular prisms with different dimensions have the same volume?
• Can rectangular prisms with the same volume have different surface areas?
• If you change the position of the rectangular prism, such as lay it down on its side, does that change its volume? Does it change the number of cubes that fills it?
• Considering your model for the volume of a rectangular prism, discuss how to find the volume of a triangular prism. How do you know? How does it relate to the volume of the rectangular prism?

Listen for . . .

• Does the student make the connection between number of cubes needed to fill the rectangular prism and the product of "length × width × height?"

Look for . . .

• Can the student connect the model to a formula to find volume?
• Does the student recognize that orientation does not affect volume?
• Does the student notice that two rectangular prisms can have the same volume but different dimensions and surface areas?

(7.9.c) Measurement. The student solves application problems involving estimation and measurement. The student is expected to estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders.

Clarifying Activity with Assessment Connections

Students consider that they have been given 360 cm³ of modeling clay and they need to design at least three different plastic containers in which to store the clay—one that is a rectangular prism, one that is a triangular prism, and one that is a cylinder. After students have had an opportunity to design their own prisms and cylinders, they then consider designing prisms and/or cylinders that will hold the same volume of clay, but have restrictions such as the following:

• A cylinder whose base has an area of 100 cm². First, record an estimate of the other dimensions of the cylinder.
• A triangular prism whose base has a length of 12.5 cm. First, record an estimate of the other dimensions of the prism.
• A rectangular prism with a height of 28 cm. First, record an estimate of the other dimensions of the prism.
• A cylinder that has a diameter of 15 cm. First, record an estimate of the other dimensions of the cylinder.

Assessment Connections

Questioning . . .

Open with . . .

• How can you design a package that will hold a certain volume of modeling clay?

Probe further with . . .

• How did you determine the dimensions to use for your package?
• If you knew the area of the base for a given package, how did you find the dimensions for that package?
• How did you use the various formulas for volume as you were working on this activity?
• In what way(s) did you use estimation to help you determine the dimensions for a given package?

Listen for . . .

• Does the student understand which dimensions are needed in order to find the volume of a particular solid?
• Does the student understand the relationship between the area of the base of a prism or cylinder and its volume?
• Is the student able to use estimation to find the dimensions for the container(s)?
• Does the student use appropriate mathematical vocabulary?

Look for . . .

• Is the student using the relationship between the area of the base for a given prism or cylinder and its volume?
• Is the student using estimation to find the dimensions for each of the containers?
• Is the student using the formulas for volume of prisms and cylinders?
• Does the student use correct units?

TAKS Connections

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

• Spring 2003, grade 7, item 21
• Spring 2004, grade 7, item 21
• Spring 2006, grade 7, item 28

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(7.10) Probability and statistics. The student recognizes that a physical or mathematical model can be used to describe the experimental and theoretical probability of real-life events.

(7.10.a) Probability and statistics. The student recognizes that a physical or mathematical model can be used to describe the experimental and theoretical probability of real-life events. The student is expected to construct sample spaces for simple or composite experiments.

Clarifying Activity with Assessment Connections

Students display, compare, and contrast the sample spaces of an event with and without replacement. For example, five students are in a drawing for two available door prizes.

Assessment Connections

Questioning . . .

Open with . . .

• How can you construct an organized list of all the possible outcomes?

Probe further with . . .

• How do you know you have listed all of the possible outcomes?
• What are all the possible outcomes if we conduct the survey without replacement?
• What are all the possible outcomes if we conduct the survey with replacement?
• Based on your sample space for each of the two scenarios (with and without replacement), what is the probability of a person winning a door prize?
• How do you know which scenario (with or without replacement) will give you a better chance of winning?
• What is the probability of a person winning both door prizes for each of the scenarios?

Listen for . . .

• Can the student justify that he or she has listed all the possible outcomes?
• Does the student have a systematic way or finding his or her sample space?
• Can the student explain the model he or she constructed?

Look for . . .

• Can the student show all the possible outcomes?
• Can the student use a variety of strategies to construct their sample space?
• Is the student's list well organized?

TAKS Connections

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

• Spring 2003, grade 7, item 24
• Spring 2004, grade 7, item 28
• Spring 2006, grade 7, item 35

(7.10.b) Probability and statistics. The student recognizes that a physical or mathematical model can be used to describe the experimental and theoretical probability of real-life events. The student is expected to find the probability of independent events.

Clarifying Activity with Assessment Connections

Students have a number cube and a two-color counter with red and yellow sides. The students throw the number cube and flip the two-color counter and record the results. The class can pool their results and then use the results to describe the approximate probability of obtaining an even roll on the number cube and a yellow on the counter.

Assessment Connections

Questioning . . .

Open with . . .

• What can you say about the probability of this event?

Probe further with . . .

• What do the results of your experiment tell you about the probability of rolling an even number and getting the yellow side of the counter?
• Can you describe how you will record the data?
• How can you determine the theoretical probability of rolling an even number and getting the yellow side of the counter?
• What are the theoretical probabilities and how do they compare to our experimental results?
• If we continued the experiment for 100 trials, how would this change the probability? (The theoretical probability will not change. The experimental probability may change. It is more likely that the experimental will be closer to the theoretical as the number of trials increases.)
• Based on your experiment, what is the experimental (or theoretical) probability of the counter landing yellow? What is the experimental (or theoretical) probability of rolling an even number?
• How do the experimental (or theoretical) probability of the counter landing yellow and the experimental (or theoretical) probability of rolling an even number relate to the experimental (or theoretical) probability of the counter landing on yellow and the number cube landing on an even number? (The experimental [or theoretical] probability of the counter landing on yellow multiplied by the experimental [or theoretical] probability of rolling an even number is the experimental [or theoretical] probability that the counter lands on yellow and the number cube lands an even number.)
• Are the events of rolling an even number and the counter landing on yellow independent events? How do you know?

Listen for . . .

• Does the student understand the concept of independent events?
• Is the student conducting the experiment correctly?
• Is the student able to approximate the probability of an event using experimental probability?
• Is the student naming the outcome correctly?
• Is the student making a generalization about the probability based on the experiment?

Look for . . .

• Can the student find probabilities for independent events?
• Is the student able to systematically record data?
• Is the student recording the probability correctly?
• Is the student conducting the experiment enough times to make a sound conclusion?
• Can the student determine the theoretical probability of the event?
• Does the student show all of the possible outcomes?

TAKS Connection

• This student expectation is not tested on TAKS. Although this is not directly tested at Grade 7, it is an important foundation for student expectations tested at later grades.

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.

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(7.11) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation.

(7.11.a) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation. The student is expected to select and use an appropriate representation for presenting and displaying relationships among collected data, including line plot, line graph, bar graph, stem and leaf plot, circle graph, and Venn diagrams, and justify the selection.

Clarifying Activity with Assessment Connections

Students collect and organize information about the amount of time they spend on days during the week and days during the weekend in different activities, e.g., eating, watching television, listening to CDs, sleeping, going to school. Students then decide on an appropriate representation for the data and justify their selection. For example, "Circle graphs are good because they show you the fraction of the day spent doing each activity and make it easy to compare the times spent on different activities."

Assessment Connections

Questioning . . .

Open with . . .

• How are you going to represent your data to answer this question?

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?

Listen for . . .

• Is the student considering the data and the context 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?
• 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 2003, grade 7, item 41
• Spring 2004, grade 7, item 40

(7.11.b) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation. The student is expected to make inferences and convincing arguments based on an analysis of given or collected data.

Clarifying Activity with Assessment Connections

Students use their graphs of time spent in various activities (described in Activity 7.11A) to compare weekend days to weekdays and write a paragraph describing how advertisers could use this information.

Assessment Connections

Questioning . . .

Open with . . .

• What inferences and arguments can you make from your data?

Probe further with . . .

• Based on your data, what activity would a student most likely do on a weekday? On a weekend?
• Which activity would most likely not occur on a weekend? On a weekday?
• How might an advertiser use this information?

Listen for . . .

• Is the student making reasonable inferences and arguments?
• Is information from student's graph reflected in his or her paragraphs?

Look for . . .

• Is the student describing his or her data correctly and making appropriate inferences?
• Is the student using language such as "most likely," "least likely," and "probable"?
• Is the student using percentages and fractions to describe his or her data?

TAKS Connections

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

• Spring 2004, grade 7, item 36
• Spring 2006, grade 7, items 20 and 46

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(7.12) Probability and statistics. The student uses measures of central tendency and range to describe a set of data.

(7.12.a) Probability and statistics. The student uses measures of central tendency and range to describe a set of data. The student is expected to describe a set of data using mean, median, mode, and range.

Clarifying Activity with Assessment Connections

Students determine the mean, median, mode, and range for a set of data. An example might include the following scenario:

The service club raised money by selling yearbook ads. Members sold ads for the following amounts: $12, $25, $12, $18, $12, $15, $12, $15, $25, $15, $18, $15, $25, $12, and $150. Describe the mean, median, mode, and range of the data and explain what each tells us about the data.

Assessment Connections

Questioning . . .

Open with . . .

• Given the data, find the measures and describe what each tells about the data.

Probe further with . . .

• How do you find the mean of the data?
• How do you find the median of the data?
• How do you find the mode of the data?
• How do you find the range of the data?
• What can you tell me about the range of the data?
• Do each of the measures tell you something different?
• What other information is needed to find these measures?
• Which will give you the price of the ad most often sold?
• Could you represent your data differently so that finding these measures might be easier?
• Can some of these measures be the same number?
• If you compare the measures of central tendencies, which would least describe the data?

Listen for . . .

• Is the student using correct language when describing the data?
• Is the student organizing the data?
• Is the student correctly summarizing the data?
• Does the student know the difference between the measures?

Look for . . .

• Does the student use a method for displaying the data to help him or her find the measures?
• Does the student use different ways to find the measures of central tendencies (for example, folding a strip of paper to find the median)?

TAKS Connections

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

• Spring 2003, grade 7, item 27
• Spring 2004, grade 7, item 29
• Spring 2006, grade 7, items 22 and 40

Extension Using Technology

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

(7.12.b) Probability and statistics. The student uses measures of central tendency and range to describe a set of data. The student is expected to choose among mean, median, mode, or range to describe a set of data and justify the choice for a particular situation.

Clarifying Activity with Assessment Connections

Students continue the task begun in Activity 7.12A. They construct a display and evaluate which of the measures—mean, median, mode, or range—best describe the set of data. Students then generate situations for which each measure of center or measure of spread could be used. (For example, if a new student wanted a general idea of how many dollars worth of ads he or she should expect to sell, which would be the most representative measure?) Additionally, the students discuss how changes in the data may affect the different measures.

Assessment Connections

Questioning . . .

Open with . . .

• Which measure would you use to describe the data and justify your choice?

Probe further with . . .

• What does each of the descriptions of the data tell you?
• Which description should you choose if you want to know which amount was sold most often?
• If the data is a representative sample of the 30 people in the class, which measure (mean, median, mode, or range) is most useful to estimate the total amount sold by the entire class? Why?
• Why might someone use one measure over another?

Listen for . . .

• Is the student using the language of mean, median, mode, and range correctly?
• Is the student recognizing the appropriateness of the measures for the given context?
• What are the strategies that the student is using to address the problem?
• Does the student recognize how changing the data will affect the measures of center or range?

Look for . . .

• Is the student organizing data efficiently to find median, mode, and range?
• Is the student using appropriate strategies to find the mean?

TAKS Connections

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

• Spring 2003, grade 7, item 12

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(7.13) Underlying processes and mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

(7.13.a) Underlying processes and mathematical tools. The student applies Grade 7 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 7.11A, Probability and statistics. Additional activities that exemplify this student expectation include 7.2B, 7.2D, 7.2.G, 7.12A.

TAKS Connections

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

• Spring 2004, grade 7, item 48
• Spring 2006, grade 7, items 6 and 17

(7.13.b) Underlying processes and mathematical tools. The student applies Grade 7 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 7.3B, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 7.2B, 7.2C, 7.2G, 7.3A, 7.5A, 7.5B, 7.9.

TAKS Connections

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

• Spring 2004, grade 7, items 6 and 8
• Spring 2006, grade 7, item 32

(7.13.c) Underlying processes and mathematical tools. The student applies Grade 7 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 7.3A, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 7.1A, 7.2A, 7.2D, 7.2F, 7.4A, 7.4B, 7.4C, 7.5A, 7.8B, 7.9.

TAKS Connections

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

• Spring 2003, grade 7, items 20 and 38
• Spring 2004, grade 7, item 15

(7.13.d) Underlying processes and mathematical tools. The student applies Grade 7 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 7.2A, Number, operation, and quantitative reasoning. Additional activities that exemplify this student expectation include 7.1B, 7.2E, 7.6A, 7.9, 7.10A, 7.11.

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(7.14) Underlying processes and mathematical tools. The student communicates about Grade 7 mathematics through informal and mathematical language, representations, and models.

(7.14.a) Underlying processes and mathematical tools. The student communicates about Grade 7 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 7.2.B, Number, operation, and quantitative reasoning. Additional activities that exemplify this student expectation include 7.1C, 7.2A, 7.2C, 7.2D, 7.2G, 7.6B.

TAKS Connections

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

• Spring 2004, grade 7, item 10
• Spring 2006, grade 7, items 19 and 47

(7.14.b) Underlying processes and mathematical tools. The student communicates about Grade 7 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 7.4C, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 7.2E, 7.2F, 7.3A, 7.4A, 7.4B, 7.6B.

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

(7.15.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 7.1C, Number operation, and quantitative reasoning. Additional activities that exemplify this TEKS include 7.1A, 7.2A, 7.2C, 7.6B, 7.6D, 7.7B, 7.8A, 7.8B.

TAKS Connections

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

• Spring 2003, grade 7, item 47
• Spring 2004, grade 7, item 13
• Spring 2006, grade 7, item 2

(7.15.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 7.4A, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 7.1C, 7.2D, 7.2F, 7.6A, 7.6B, 7.6C, 7.6D, 7.7A, 7.7B, 7.8A, 7.8B, 7.9.

TAKS Connections

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

• Spring 2004, grade 7, item 41
• Spring 2006, grade 7, items 27 and 45

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