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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.

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

(6.1.a) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to compare and order non-negative rational numbers.

Clarifying Activity with Assessment Connections

The teacher posts an unlabeled, classroom-size number line marked with the benchmarks 0, 1/2, and 1. The teacher provides students with a variety of non-negative rational numbers (fractional and decimal form) written on note cards. Students put the note cards in the appropriate places on the number line. Download number cards here (pdf 268 kb).

Assessment Connections

Questioning . . .

Open with . . .

• Explain how you can locate these numbers on a number line.

Probe further with . . .

• What strategies can you use to locate these numbers?
• Which number has the greatest value? How do you know?
• Which number has the least value? How do you know?
• How do you know when a number is more than 1/2 or less than 1/2?
• Is your number closer to 0 or 1/2?
• What does it mean when two numbers share the same spot on the number line?
• Is your number equivalent to any other number on the number line?

Listen for . . .

• Can the student verbalize the strategies used?
• What strategies does the student use when comparing the numerator to the denominator (such as part-to-whole)?
• Can the student use appropriate vocabulary to describe the relationship between two rational numbers?
• Can the student express rational numbers in a variety of equivalent forms (such as both decimals and fractions)?

Look for . . .

• Can the student generate equivalent fraction and decimal forms?
• Can the student use benchmarks to position numbers on the number line?
• Can the student place the rational number in the appropriate range on the number line?
• Can the student compare and order non-negative rational numbers?

TAKS Connections

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

• Spring 2003, grade 6, item 16
• Spring 2004, grade 6, item 31
• Spring 2006, grade 6, item 32

(6.1.b) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to generate equivalent forms of rational numbers including whole numbers, fractions, and decimals.

Clarifying Activity with Assessment Connections

The teacher prepares a jar of fifty marbles of different colors. Pairs of students draw a handful of marbles from the jar and record the numbers of each color and total number of marbles drawn. They return the marbles, pass the jar to the next pair, and then generate a variety of ratios that can describe the handful of marbles. Students then determine equivalent forms of the rational numbers that they generated.

For example, a jar has 14 red, 28 yellow, and 8 white marbles. Students pull 18 marbles from the jar: 6 red, 9 yellow, and 3 white. They may describe their handful of marbles in the following way:

The portion of the handful that was red is 6/18 = 3/9 = 1/3 = 9/27 = 25/75.

The portion of the handful that was yellow is 9/18 = 1/2 = 50/100 = 3/6 = 50% = 0.5.

The portion of the handful that was white is 3/18 = 1/6 = 6/36 = 10/60.

Assessment Connections

Questioning . . .

Open with . . .

• How can you represent the handful of marbles as a fraction and a decimal?

Probe further with . . .

• What information do you need to know to describe the relationship of the marbles?
• What fractional part have you drawn? Are there other ways to describe this fractional part? What are they?
• What decimal can you use to describe the relationship of the marbles?
• How do you find decimal equivalents?
• What strategies did you use to create decimal equivalents?
• How can you check to see if your strategy worked?
• How did your knowledge of equivalent fractions help you determine equivalent decimals?

Listen for . . .

• Can the student verbalize the strategies used?
• Is the student using grade-appropriate language to describe fractions and decimals?
• Can the student describe a process for generating equivalent forms of rational numbers?
• Can the student explain his or her reasoning?

Look for . . .

• Can the student write a ratio that describes the handful of marbles to the total number of marbles in the jar?
• Can the student write other ratios to describe other relationships between the marbles?
• Can the student generate equivalent forms for a rational number?
• Does the student have a strategy for generating equivalent forms for a rational number?
• Can the student use fractions to generate decimals and other fractions, and decimals to generate fractions and other decimals?

TAKS Connections

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

• Spring 2003, grade 6, item 20
• Spring 2004, grade 6, item 28
• Spring 2006, grade 6, item 34

(6.1.c) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to use integers to represent real-life situations.

Clarifying Activity with Assessment Connections

The teacher describes the scenario of a football player running 20 yards down the field past the line of scrimmage. Students provide words that a sports commentator could use to describe this play. The teacher highlights the words or phrases that signify a positive gain.

Next, the teacher describes a scenario of a football player being sacked 20 yards behind the line of scrimmage. Students provide words that a sports commentator could use to describe this play. The teacher highlights the words or phrases that signify a loss.

Through conversation, the teacher guides the students in making the connection between their generated phrases and the symbolic representation of positive/negative numbers.

Given the following real-life situations, the students determine an integer that best describes the results. Students should provide explanations for their choices.

• The quarterback is sacked 5 yards behind the line of scrimmage.
• The temperature is 70 degrees in the morning and rises to 90 degrees in the afternoon.
• A golfer completes a hole 2 under par on a par 4 hole.
• A submarine is 300 feet below the surface of the ocean.
• The temperature is 10 degrees and falls 25 degrees.
• A game show contestant has 100 points and incorrectly answers a 500-point question.
• A middle school has 310 sixth-graders, and 10 sixth-graders move to another school.

Assessment Connections

Questioning . . .

Open with . . .

• What situations represent a loss? How do you know when a number is negative? Positive?

Probe further with . . .

• What kind of situation would result in zero?
• What vocabulary did you use to describe each situation in this activity?
• How did you symbolize that vocabulary?
• What kind of number sentence would describe a series of football plays? A series of submarine maneuvers?

Listen for . . .

• Can the student identify a situation that would result in zero?
• Does the student recognize that subtraction does not necessarily result in a negative number?
• Can the student describe integers in terms of temperature readings, football yardage, loss of points, etc.?
• Can the student explain his or her thought process?
• Can the student generalize situations as positive or negative?

Look for . . .

• Does the student use correct symbolic representation?
• Does the student model the situation correctly?
• Can the student use several approaches to justify his or her response?

TAKS Connections

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

• Spring 2003, grade 6, item 36
• Spring 2006, grade 6, item 14

(6.1.d) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to write prime factorizations using exponents.

Clarifying Activity with Assessment Connections

In a number search puzzle like the one below, students draw a circle around combinations of numbers that are factors of 360. Students list each string of numbers and compare the factors. For example, the string 9 x 40 should be written 9 x 40 = 360.

Students can name the strings with two factors, three factors, four factors, and so on, and then discuss what they notice about the strings of factors. They rewrite the strings of factors using exponents. For example, 45 x 2 x 2 x 2 = 45 x 2³. Students note where the longest string is located and what it is.

Next, students use the factor strings to create factor trees for 360.

Assessment Connections

Questioning . . .

Open with . . .

• What are the factor strings for 360?

Probe further with . . .

• Is this factor string (for example, 45 x 23) a prime factorization for 360? Why or why not?
• What is the longest string of factors you can find for 360?
• How can you use the factor trees to find prime factorization?
• Is there another way to express prime factorization?
• If you start with a factor pair, can you factor those numbers?
• How do know when you have found the longest string of factors?
• If you start with a factor pair, can you factor those numbers and rewrite them using exponents?

Listen for . . .

• Can the student list the factors of the number?
• Does the student clearly verbalize observations?
• Does the student identify more than one strategy to find factors?
• Does the student use one string of factors to determine additional factor strings?
• Does the student recognize when a number is prime or composite?

Look for . . .

• Has the student found the longest string?
• Can the student write prime factorizations in exponential notation?
• Can the student determine the difference between composite and prime numbers?
• Are students making connections between exponential notation and the numbers they represent?

TAKS Connections

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

• Spring 2003, grade 6, item 14
• Spring 2004, grade 6, item 39
• Spring 2006, grade 6, item 3

(6.1.e) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to identify factors of a positive integer, common factors, and the greatest common factor of a set of positive integers.

Clarifying Activity with Assessment Connections

Two pairs of students play a greatest common factor game using a set of index cards numbered 1 through 32. Students shuffle the cards well. They turn over the first 16 cards and display them face up on the table top, leaving the remaining 16 cards in the deck. In the first round of play, Pair 1 pull a number card from the remaining deck. They then select a card from the table. The pair earn points for this round equivalent to the greatest common factor of the two cards. Pair 1 record their score for this round and keep the two cards.

For example: Pair 1 pull 21 from the deck and decide to select 14 from the table for a score of 7 for this round. They keep cards 21 and 14, so these cards can no longer be used in this game.

Pair 2 then pull a number card from those remaining in the deck, select a card from the table, and record the score for this round, again keeping the two cards.

For example: Pair 2 take 17 at random from the deck. This is bad luck, since 17 is relatively prime with respect to the numbers 1-32. They choose 23, which is on the table for a score of 1 for this round. Pair 2 keep cards 17 and 23 so these cards can no longer be used in this game.

At each turn, the pair select two number cards, one from the deck and one from the table. When there are no more cards available, the game is over. Each player pair totals their scores from all rounds and the pair with the greatest sum wins.

Assessment Connections

Questioning . . .

Open with . . .

• What strategies did you use to play the game?

Probe further with . . .

• What is a factor? What is a multiple?
• Why did you choose that number card from the table?
• How did you decide which card you wanted to pick up? Why?
• How did you find your score for the round?
• How did you determine the greatest common factor?
• How did you determine the factors for the numbers? Do you have a strategy?
• What directions and advice would you give your friend who is going to play this game for the very first time?

Listen for . . .

• Can the student distinguish between factors and multiples?
• Do the students notice the difference between factors in general and the greatest common factor?
• Can the student verbalize a strategy for finding common factors and the greatest common factor?
• Does the student use appropriate vocabulary when describing his or her strategy to play the game?

Look for . . .

• Can the student determine factors of a number?
• Does the student select the number card to find the greatest common factor available?
• Does the student recognize an effective strategy for winning the game?

TAKS Connections

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

• Spring 2004, grade 6, item 17
• Spring 2006, grade 6, item 15

(6.1.f) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to identify multiples of a positive integer and common multiples and the least common multiple of a set of positive integers.

Clarifying Activity with Assessment Connections

Two pairs of students play a lowest common multiple game using a set of index cards numbered 1 through 32. Students shuffle the cards well. They turn over the first 16 cards and display them face up on the table top, leaving the remaining 16 cards in the deck. In the first round of play, Pair 1 pull a number card from the remaining deck. They then select a card from the table. The pair earn points for this round equivalent to the least common multiple of the two cards. Pair 1 record their score for this round and keep the two cards.

For example: Pair 1 pull 21 from the deck and decide to select 14 from the table for a score of 42 for this round. They keep cards 21 and 14, so these cards can no longer be used in this game.

Pair 2 then pull a number card from those remaining in the deck, select a card from the table, and record the score for this round, again keeping the two cards.

For example: Pair 2 pull 1 from the deck. This is good luck. They choose 3, which is on the table for a score of 3 for this round. Pair 2 keep cards 3 and 1, so these cards can no longer be used in this game.

At each turn, the player pair select two number cards, one from the deck and one from the table. When there are no more cards available, the game is over. Each player pair total their scores from all rounds and the pair with the least sum wins.

Assessment Connections

Questioning . . .

Open with . . .

• What strategies did you use to play the game?

Probe further with . . .

• What is a factor? What is a multiple?
• Why did you choose that number card from the table?
• How did you decide which card you wanted to pick up? Why?
• How did you find your score for the round?
• How did you determine the least common multiple?
• How did you determine the factors for the numbers? Do you have a strategy?
• What directions and advice would you give your friend who is going to play this game for the very first time?

Listen for . . .

• Can the student distinguish between factors and multiples?
• Do the students notice the difference between multiples in general and the least common multiple?
• Can the student verbalize a strategy for finding the least common multiple?
• Does the student use appropriate vocabulary when describing his or her strategy to play the game?

Look for . . .

• Can the student determine multiples of a number?
• Does the student select the number card to find the least common multiple available?
• Does the student recognize an effective strategy for winning the game?

TAKS Connections

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

• Spring 2003, grade 6, item 38
• Spring 2004, grade 6, item 29

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

(6.2.a) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions. The student is expected to model addition and subtraction situations involving fractions with objects, pictures, words, and numbers.

Clarifying Activity with Assessment Connections

Students use the drawing below to name the fractional parts of the whole, then find shapes or combinations of shapes that are equivalent to each other. Students write statements that show equivalence and then write the numerical representations for each statement. Models, words, and fractional representations are all used to prove the statements. For example, A = B + C or 1/4 = 1/8 + 1/8 or C - E = D.

Assessment Connections

Questioning . . .

Open with . . .

• How can you model addition and subtraction of fractions using the area model?

Probe further with . . .

• How do you name each fractional part?
• Are there any benchmark fractional parts? (For example, the square is divided into fourths.)
• What is the largest fractional piece of the square?
• What is the smallest? How do you know?
• How many different pieces are there in the square? Does that help you determine each fractional part?
• Does each piece represent the same fractional part?
• What might you do to make naming the fractional parts easier?
• Can you show two or more parts of the square that are equivalent to another part of the square?
• Do these parts have to be the same shape?
• What operation did you complete to show that the parts were equal?
• Can you model an addition situation using the parts of the square? Can you write this as a statement? Can you justify why this works?
• Can you model a subtraction situation using the parts of the square? Can you write this as a statement? Can you justify why this works?

Listen for . . .

• Does the student use the correct fractional representation when referring to each piece?
• Does the student use correct mathematical terminology?
• Does the student use equivalency to justify naming of fractional parts?

Look for . . .

• Does the student consider the whole square when referring to the fractional parts?
• Does the student have a sense of size when referring to fractional representations?
• Does the student recognize that all the fractional parts of the square together are equal to the whole?
• Can the student manipulate the fractional pieces to determine equivalencies?
• Can the student rename fractional parts?
• Does the student make a connection between the fractional representation and each piece? Can the student name them?
• Can the student create an equation using fractions?
• Can the student see that two or more fractional parts are equivalent to a larger fractional part?
• Does the student compare fractions correctly?

TAKS Connections

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

• Spring 2003, grade 6, item 12
• Spring 2004, grade 6, item 11
• Spring 2006, grade 6, item 29

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

Clarifying Activity with Assessment Connections

Using a menu, recipes, or an ad for a grocery store, students will create problem situations involving addition and subtraction of fractions and decimals. Students will then exchange problems and answer the problems written by others.

For example, Jack bought a hamburger, fries, and a medium soda. How much did Jack spend before tax and tip? If Jack added $1.50 for tax and tip and paid with a $20.00 bill, how much change did he receive? Or, if Recipe 1 calls for 1/4 teaspoon of salt and Recipe 2 calls for 3/8 teaspoon of salt, how much more salt is required by Recipe 2? Or, Yvonne wants to by 1/2 pound of cheese and Will needs 3/4 of a pound, how much do they need to buy?

Assessment Connections

Questioning . . .

Open with . . .

• How can you create addition and subtraction problems from a menu, recipes, or ad?

Probe further with . . .

• How much is a pound of ______?
• How much is a half of a pound of ______?
• How much is a quarter of a pound of ______?
• How much would you have to pay for two ______?
• How do you know what operation to use?
• How do you add fractions? Subtract fractions?
• How do you add decimals? Subtract decimals?

Listen for . . .

• Does the student recognize when a problem is an addition situation?
• Does the student recognize when a problem is a subtraction situation?
• Does the student rename fractions correctly to solve the equations?
• Does the student work with appropriate place values?
• Does the student verbalize the need to have common denominators to add and subtract?
• Does the student have a method for finding common denominators?

Look for . . .

• Can the student write a problem with multiple steps?
• Can the student represent equivalent fractions in more than one way?
• Can the student use a common denominator?
• Does the student assess the problem and formulate a strategy before starting the problem?
• Does the student recognize fractional parts of amounts?
• Does the student consider reasonableness when checking the answer?

TAKS Connections

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

• Spring 2003, grade 6, item 42
• Spring 2004, grade 6, item 21
• Spring 2006, grade 6, item 21

(6.2.c) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions. The student is expected to use multiplication and division of whole numbers to solve problems including situations involving equivalent ratios and rates.

Clarifying Activity with Assessment Connections

Students pretend that they are shopping for CDs and DVDs. The store sells CDs and DVDs using the following pricing:

• 3 CDs for $24
• 2 DVDs for $30

Each student has $65 to spend on CDs and DVDs. What combinations of CDs and DVDs could they purchase? While students are determining purchase combinations, the teacher announces that they each discover another $19 in their pockets. What are the new combinations of CDs and DVDs could students now purchase?

Assessment Connections

Questioning . . .

Open with . . .

• What strategy would you use to determine your purchase?

Probe further with . . .

• If you bought only CDs, how many could you purchase?
• If you bought only DVDs, how many could you purchase?
• If you want at least one CD and one DVD, what purchase combinations could you make?
• How much are you paying for one CD when you purchase 3 for $24?
• What factors or multiples do you have to consider when making your purchase?

Listen for . . .

• Does the student adequately explain and justify the purchases?
• Does the student develop a pattern to determine solutions?
• Does the student consistently use the rates given in determining the purchases?
• Is the student accurately using unit rates to determine combination possibilities?

Look for . . .

• Does the student explore more than one solution?
• Is the student using multiplication and division to determine answers (versus addition and subtraction)?

TAKS Connections

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

• Spring 2004, grade 6, item 15
• Spring 2006, grade 6, items 4 and 11

(6.2.d) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions. The student is expected to estimate and round to approximate reasonable results and to solve problems where exact answers are not required.

Clarifying Activity with Assessment Connections

The math club runs a store that sells school supplies. Listed below are the items they sell.

Students imagine that they are waiting in line to restock their supplies. They want to spend between $1.50 and $2.00, and they have to quickly decide what they want because the line is moving fast. What are some combinations of items they could buy?

Assessment Connections

Questioning . . .

Open with . . .

• What combination of items can you buy? How did you determine what you can buy?

Probe further with . . .

• Do you need to know an exact dollar amount?
• What price would you round the pencils to?
• What mathematics can you apply to the situation to quickly come up with a combination of items to buy?
• What will happen if your rounded number is greater than the original price? Less than the original price?
• What will happen if you round all prices before making your decision?
• What combination can you buy if you only want pencils and paper?
• What combination can you buy if you have $10?
• What combination can you buy if you need at least two folders?

Listen for . . .

• Does the student round prices?
• Does the student verbalize strategies for rounding?
• Does the student discuss approximate answers or exact answers?
• Does the student consider rounded prices when determining cost?

Look for . . .

• Does the student assume 30¢ to 35¢ for the price of a pen?
• Does the student round all amounts?
• Does the student adjust answers based on the rounded prices?
• Can the student justify that his or her combinations work?

TAKS Connections

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

• Spring 2003, grade 6, item 44
• Spring 2004, grade 6, item 5
• Spring 2006, grade 6, item 17

(6.2.e) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions. The student is expected to use order of operations to simplify whole number expressions (without exponents) in problem solving situations.

Clarifying Activity with Assessment Connections

Two pairs of students play "Make a Dozen" using a set of 12 index cards numbered 1 through 12. Students shuffle the cards of the deck. In the first round of play, Pair 1 select 4 cards from the random deck and display them on the table. They use those numbers along with the operations of multiplication, division, addition, subtraction, and parentheses to create an algebraic expression whose value is 12. They write the number sentence and return the cards to the deck. Pair 2 check the number sentence for accuracy.

For example, Pair 1 may have selected 5, 7, 8, and 3. They could write the number sentence (7 - 3)*(8 - 5) = 12.

Then Pair 2 reshuffles the deck, selects four cards, and creates a number sentence for their numbers.

At each turn, a pair select four number cards from shuffled deck of 12 cards. They use the cards with arithmetic operations to create an algebraic expression whose value is 12 and then write the corresponding number sentence. The other player pair verify the number sentence. If the number sentence is correct, the pair that wrote the question earn one point. The first pair with three points win the game and are allowed to choose a new value in place of the 12 for the second game.

Assessment Connections

Questioning . . .

Open with . . .

• What strategies did you use to play the game?

Probe further with . . .

• What strategies did you use to develop the number sentence?
• Do you need parentheses? Why?
• In what order do you perform the operations in the number sentence?
• What operation do you perform first? What next? How do you know?
• Is there another expression that is equivalent to 12 that you could have used?

Listen for . . .

• Does the student apply proper order of operations?
• Can the student verbalize a strategy for finding an expression equivalent to 12?
• Does the student use appropriate vocabulary when describing his or her strategy to play the game?

Look for . . .

• Does the student know the order to perform operations in the number sentence developed?
• Can the student write a number sentence made with the 4 numbers the pair selected?
• Does the student use parentheses appropriately?
• Does the student recognize an effective strategy for creating the algebraic expression equal to 12?

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

(6.3.a) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to use ratios to describe proportional situations.

Note: Proportional reasoning is the "big idea" in middle school mathematics. Research suggests that the development of this concept can be supported "through exploring proportional (and nonproportional) situations in a variety of problem contexts using concrete materials or situations in which students collect data, build tables, and determine the relationships between the number pairs (ratios) in the tables." [National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Page 244.]

Clarifying Activity with Assessment Connections

Students build models of cubes using marshmallows and toothpicks (clay and pipe cleaners or similar items will also work). Students create a table to record the materials needed to build a certain number of cubes so that they can determine the relationship between the needed materials and number of cubes built.

For example, it takes 12 toothpicks and 8 marshmallows to build one cube, and 24 toothpicks and 16 marshmallows to build two cubes. The ratio of number of edges to number of vertices is 12 to 8 or 24 to 16. Students can also use ratios to describe other relationships in the cubes such as the ratio of faces to cubes (6 to 1 and 12 to 2) or the ratio of faces to vertices (6 to 8 and 12 to 16).

Assessment Connections

Questioning . . .

Open with . . .

• How can you use ratios to describe the relationships in this table?

Probe further with . . .

• How many faces are there on your cubes?
• How many vertices are there on your cubes?
• How are ratios written?
• What is meant by a proportional situation?
• How do you use ratios to express proportional situations?
• What relationship do ratios have in a proportion?
• If the ratio of toothpicks to marshmallows is 12 to 8, what meaning can you attach to the ratio 3 to 2?
• Can you describe another situation that has a ratio of 3 to 2? (For example, 3 eggs to 2 omelets, or 3 wheels to 2 pedals on a tricycle.) What is the ratio of edges to faces? Vertices to faces?

Listen for . . .

• Can the student verbalize a strategy for setting up a proportional relationship?
• Can the student describe a ratio?
• Can a student distinguish between part-to-part and part-to-whole ratios?
• Can the student describe proportional relationships?
• Is the student verbalizing the operations used?
• Does the student use mental math?
• Does the student check for reasonableness?

Look for . . .

• How does the student organize his or her information and calculations?
• Does the student label the quantities in the ratio?
• Does the student represent the proportional situation symbolically?

TAKS Connections

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

• Spring 2003, grade 6, item 24
• Spring 2004, grade 6, item 26
• Spring 2006, grade 6, item 7

(6.3.b) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to represent ratios and percents with concrete models, fractions, and decimals.

Clarifying Activity with Assessment Connections

Given the following ratios and percentages, students should represent or model them using paper strips of equal size or percent grids:

1. 50% of the sixth-graders at our school like broccoli.
2. 9 out of 10 teenagers wear jeans to school.
3. The ratio of the number of students who do not take the bus to school to the number of those who do is 6 to 4.
4. 3 out of 4 professional athletes played college sports.

Students can fold, color, or cut the paper strips or percent grids to make their representations. After making the representations, students should write the fraction and decimal for each situation.

Assessment Connections

Questioning . . .

Open with . . .

• How do you represent or model these examples with paper strips?

Probe further with . . .

• What does 100% of a paper strip look like?
• What fractions can be used to describe the information in the examples?
• What decimals can be used to describe the information in the examples?
• What other percentages or ratios can you represent with the paper strips?
• What percentages or ratios can you identify using objects in the classroom (such as ceiling tiles, desks, etc.)

Listen for . . .

• Does the student understand that one paper strip represents 100% or the whole?
• Does the student correctly identify concrete examples of percentages or ratios in the classroom?
• Is the student correctly describing the ratios and percentages as decimals and fractions?

Look for . . .

• Does the student correctly fold, shade, or cut the paper strips?
• Does the student check with other students to see if the strips are congruent?

TAKS Connections

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

• Spring 2004, grade 6, item 14
• Spring 2006, grade 6, items 16 and 18

(6.3.c) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to use ratios to make predictions in proportional situations.

Clarifying Activity with Assessment Connections

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

You are part of the committee to organize a school carnival for preschoolers. You must create a spinner game that will let most people win one of three different small prizes but only a few people win one of the large prizes. You have a gross—that is, 12 dozen—of each of the different small prizes and 24 of the large prizes. After all the prizes are gone, you must close the booth. Design a spinner that will allow you to keep the booth open as long as possible.

Assessment Connections

Questioning . . .

Open with . . .

• In order to keep the booth open for as long as possible, how would you design the spinner?

Probe further with . . .

• What ratios can you write from the given situation?
• Based on your spinner design, if 100 people play, how many people are likely to win a small prize? A large prize?
• If 200 people play, how many are likely to win a small prize? A large prize? How will you use your ratios to make the predictions?
• If 1,000 people play, how many are likely to win a small prize? A large prize? Nothing at all?
• Compare your design to others in your class. Which spinner would you like to see at the carnival? Why?
• What is the chance (ratio) for winning the large prizes? How did you determine your ratio?
• How did you decide on your spinner design?

Listen for . . .

• Is the student accurately describing the part-to-whole relationships from the spinner?
• Is the student using an effective strategy to make a prediction?

Look for . . .

• Is the student creating proportional relationships from the spinner to make predictions?
• Can the student write correct ratios?
• Does the student divide the spinner appropriately?
• Does the student use correct part-to-part or part-to-whole ratios?
• Does the student's solution satisfy the conditions of the problem?

TAKS Connections

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

• Spring 2003, grade 6, item 10
• Spring 2004, grade 6, item 37
• Spring 2006, grade 6, item 26

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(6.4) Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes.

(6.4.a) Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes. The student is expected to use tables and symbols to represent and describe proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter and area.

Clarifying Activity with Assessment Connections

Students measure the side lengths and perimeters of several different-sized squares and create a table of the measurements. For example:

Students then make a list of relationships they observe between the side lengths and perimeters of the squares.

Assessment Connections

Questioning . . .

Open with . . .

• Describe any relationships between the side length of a square and its perimeter.

Probe further with . . .

• What patterns do you see in the table?
• What are the properties of the square?
• What is perimeter?
• How do you find perimeter?
• How does the length of the sides affect the perimeter?
• How can you describe the relationship between the length of the sides and perimeter?
• What ratios do you see in this table? (For example, the ratio of the length of a side to its perimeter is 1 unit to 4 units or 1/4; the ratio of perimeter to the length of a side is 4 to 1 or 4/1.)
• Can you generalize to other quadrilaterals? To other polygons?
• If you know the perimeter of a square, how can you find the length of one side?
• If you know the length of a side of a square, how can you find the perimeter?
• What equation can be used to determine p, the perimeter of a square with length of side l?
• If a square has a perimeter of 56 inches, what is the length of one side? How do you know?
• What equation can be used to determine l, the length of a side of a square with perimeter p?
• If a square has a side length of 7 inches, what is its perimeter? How do you know?

Listen for . . .

• Can the student identify perimeter?
• Does the student accurately describe the patterns?
• Can the student describe ratios in the table?
• Can the student make a conjecture about the relationship between the length of a square's side and its perimeter?

Look for . . .

• Does the student find patterns in the table?
• Does the student label the table appropriately?
• Does the student use appropriate measurements?
• Can the student use a pattern to generate table entries?

TAKS Connections

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

• Spring 2003, grade 6, item 28
• Spring 2006, grade 6, item 25

(6.4.b) Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes. The student is expected to use tables of data to generate formulas representing relationships involving perimeter, area, volume of a rectangular prism, etc.

Clarifying Activity with Assessment Connections

Students use cubes to build rectangular prisms and then record in a table the dimensions of each prism and the number of cubes used. The table should include the following information:

Students should identify the patterns they see in the table and generate formulas to represent the relationships of the prisms' characteristics.

Assessment Connections

Questioning . . .

Open with . . .

• If you know the dimensions of a rectangular prism, can you predict the number of cubes needed to build it?

Probe further with . . .

• What patterns do you notice in the rows of the table?
• Is there a relationship between the area of the base and the number of cubes?
• Is there a relationship between the height and the number of cubes?
• How many cubes are in a single layer of the rectangular prism?
• How does the number of cubes used to build a rectangular prism affect the prism's dimensions?
• How can you find the area of the base?
• How can you describe in words how you would determine the area of the base given its length and width?
• What formula can be used to determine the area of the base of the rectangular prism given the length and width?

Listen for . . .

• Is the student recognizing the relationship between area of the base and the length/width of the prism?
• Is the student recognizing the relationship between the area of the base, the height, and the number of cubes?
• Is the student using patterns to correctly predict the number of cubes needed to build additional rectangular prisms? (For example, if a rectangular prism has a base with length of 4 cubes, width of 3 cubes, and height of 6 cubes, can the student predict the number of cubes needed to build the prism?)

Look for . . .

• Can the student build a prism?
• Can the student identify the dimensions of a rectangular prism?
• Does the student correctly enter data into the table?
• Does the student know how to find the area of the base of the prism?
• Does the student recognize that the prism has layers?

TAKS Connections

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

• Spring 2003, grade 6, item 37
• Spring 2004, grade 6, item 25
• Spring 2006, grade 6, item 31

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(6.5) Patterns, relationships, and algebraic thinking. The student uses letters to represent an unknown in an equation.

(6.5) Patterns, relationships, and algebraic thinking. The student uses letters to represent an unknown in an equation. The student is expected to formulate equations from problem situations described by linear relationships.

Clarifying Activity with Assessment Connections

To help prepare for Family Math Nights at an elementary feeder campus, sixth graders are writing clues to challenge families to determine "What's Inside?" Film canisters are filled with pennies and nickels. Then students write clues indicating relationships between the number of coins and the types of coins in the canister. For example, given a film canister with a collection of 8 pennies and 3 nickels, students may create clues that describe relationships:

• There are two more than twice as many pennies as there are nickels.
• There are a total of 11 coins.
• There is one less than three times as many pennies as nickels.

At the Family Math Night, families read the clues to figure out how many pennies and nickels are inside the canister.

To challenge themselves, sixth-grade students exchange the written clue descriptions and formulate equations explaining the relationship described. For example,

• There are two more than twice as many pennies as there are nickels. Let p = number of pennies and n = number of nickels: 2n + 2 = p.
• There are a total of 11 coins. Let p = number of pennies and n = number of nickels: n + p = 11.
• There is one less than three times as many pennies as nickels. Let p = number of pennies and n = number of nickels: 3n - 1 = p.

Assessment Connections

Questioning (before exchanging descriptions) . . .

Open with . . .

• Tell me about a relationship between the number of pennies and the number of nickels that you used for a clue.

Probe further with . . .

• What did you think about when you were looking for a relationship?
• Have you provided all of the needed information? What information is needed to formulate an equation describing this relationship?

Listen for . . .

• Does the student's explanation match his or her written work?

Look for . . .

• Can the student describe a relationship that matches the given numbers?
• Does the student provide information needed to formulate the equation to solve the problem?
• Does the student describe the relationship in a way that is clear and easily interpreted?

Questioning (after exchanging descriptions) . . .

Open with . . .

• What equation can you generate matching the clue description?

Probe further with . . .

• What information is given?
• What information is needed to formulate the equation?
• Can you draw a picture or write a symbolic representation for the problem?
• What do the letters in your equation represent?
• Does your equation seem reasonable?

Listen for . . .

• Does the student's explanation match his or her written work?
• Is the student able to discuss the reasonableness of his or her equation?

Look for . . .

• Can the student formulate an equation?
• Does the student explain the meaning of the symbols used in the equation?
• Can the student identify the information necessary to form the equation?
• Does the equation match the student's explanation?
• Can the student self-correct any errors?

TAKS Connections

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

• Spring 2004, grade 6, item 19
• Spring 2006, grade 6, item 36

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(6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.

(6.6.a) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to use angle measurements to classify angles as acute, obtuse, or right.

Clarifying Activity with Assessment Connections

Students measure angles found in objects around the room and label the angles with the appropriate classification (acute, obtuse, or right).

Assessment Connections

Questioning . . .

Open with . . .

• What determined the label you chose for each angle?

Probe further with . . .

• How many categories did you use?
• What makes an angle acute? Right? Obtuse?
• How did you use visual clues to identify each angle?
• Did you find examples for all angle classifications?
• Can an angle be in more than one category? Why?
• What helped you categorize the angles accurately?
• Is one type of angle represented more often than another type of angle? Or less? Why?
• Is there a connection between an object's angles and its shape? Can you justify your answer?
• Do the object's angles relate to the design of the structure? If so, how?
• What do the angles tell us about the object's function?
• How can categorizing angles help you estimate angle measurement in the real world?

Listen for . . .

• Does the student use definitions for acute, obtuse, and right angles correctly?
• Can the student justify his or her choice of labels?
• Is the student using visual clues to categorize the angles before measuring?

Look for . . .

• Can the student find angles in objects?
• Can the student use tools appropriately to measure angles?
• Can the student use organizational models to record and sort angles?
• Can the student identify the rays of the angles?

TAKS Connections

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

• Spring 2003, grade 6, item 4
• Spring 2006, grade 6, items 41 and 44

(6.6.b) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to identify relationships involving angles in triangles and quadrilaterals.

Clarifying Activity with Assessment Connections

Given a set of different triangles or quadrilaterals, students measure the angles (using a protractor or geometry exploration software) and classify polygons according to their properties. Students organize their data in a table and analyze patterns to match triangles or quadrilaterals with the most appropriate polygon name.

Assessment Connections

Questioning . . .

Open with . . .

• Describe the relationship between the angle measurements and the specific name of the triangle or quadrilateral.

Probe further with . . .

• Do you see patterns in the angle measurements of a triangle? Quadrilateral?
• Do you see similarities and differences between the polygons and their angles?
• Can the definitions of acute, obtuse, or right angles help you sort the polygons by name?
• Do any triangles or quadrilaterals have two congruent angles? Three congruent angles? All congruent angles?
• Are there ways to organize your data to help you determine patterns?

Listen for . . .

• Is the student using geometric terms to analyze the properties of each polygon?
• Does the student use the definitions for acute, obtuse, and right angles correctly?
• Is the student making conjectures about the relationships between angles and polygons?
• Is the student comparing the angle measurements?
• Is the student generalizing accurately concerning the attributes of the different types of triangles? Quadrilaterals?
• Is the student extending the angle-to-polygon analysis to specific triangle names? Specific quadrilateral names?

Look for . . .

• Is the student measuring the angles accurately?
• Is the student looking for patterns?
• Is the student representing the sorted polygons with the best geometric labels?
• Can the student categorize the polygons according to the attributes seen?
• Can the student justify the categories using specific properties of angles?
• Is the student checking polygon arrangements for errors or duplications?
• Can the student extend angle-to-polygon relationships from one category to another?
• Can the student organize his or her information in a table?

TAKS Connections

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

• Spring 2003, grade 6, item 30
• Spring 2006, grade 6, items 9 and 38

(6.6.c) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to describe the relationship between radius, diameter, and circumference of a circle.

Clarifying Activity with Assessment Connections

Students use tape measures or string to measure the radii, diameters, and circumferences of several circular objects, organize their data in a table, and look for patterns and relationships using a calculator. Students answer questions such as "What is the relationship between the length of the radius and the length of the diameter? The length of the diameter and the circumference?" (For example, "The length of the diameter is twice the length of the radius.")

Students then use other circular objects to test the patterns and relationships found. For example, students could measure circumference of a circular object to be, for instance, 12.5 inches, and then use the relationships they found to calculate the diameter. Then students measure the diameter and compare the measurement with their calculation.

Assessment Connections

Questioning . . .

Open with . . .

• What are the relationships between radius, diameter, and circumference?

Probe further with . . .

• How can you find the circumference of a circle?
• How can you find the diameter of a circle if you know the radius?
• How can you find the radius of a circle if you know the diameter?
• Do you see any patterns in the measurements that might help you predict radius, diameter, and circumference?
• If you know the diameter, how can you find the circumference?
• If you know the circumference, how can you find the diameter?
• How did your calculation of the diameter for the circle with the circumference that you found compare with the measurement you made of the diameter?

Listen for . . .

• Can the student verbalize patterns?
• Is the student making reasonable connections between the radius, diameter, and circumference of a circle?
• Does the student use appropriate vocabulary?
• Can the student hypothesize relationships?
• Does the student use the calculator correctly?

Look for . . .

• Is the student using ratios to represent relationships?
• Is the student measuring radius, diameter, and circumference with matching units of measure?
• Does the student use a unit of measure appropriate for the size of the object (e.g., feet for large objects and inches for small)?
• Is the student looking for multiplicative patterns?
• Is the student organizing his or her data and looking for patterns?
• Is the student recording his or her findings?
• Is the student using different strategies to measure?
• Is the student checking for reasonableness of calculations?

TAKS Connections

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

• Spring 2003, grade 6, item 27
• Spring 2004, grade 6, item 23
• Spring 2006, grade 6, item 13

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(6.7) Geometry and spatial reasoning. The student uses coordinate geometry to identify location in two dimensions.

(6.7) Geometry and spatial reasoning. The student uses coordinate geometry to identify location in two dimensions. The student is expected to locate and name points on a coordinate plane using ordered pairs of non-negative rational numbers.

Clarifying Activity with Assessment Connections

Students play a hide and seek game in groups of three. One student "hides" and the other two "seek" geometric figures in a coordinate system. The hider and the seekers are provided identical grids. During play, the hider is back-to-back with the seekers. The hider draws a geometric shape on the grid, making sure its vertices have integer coordinates. The hider begins the game by telling the seekers what type of geometric shape was drawn.

The goal of the game is for the seekers to cooperatively locate the points corresponding to the vertices of the hidden shape. The hider provides clues after each guess, telling the seekers where the guess is relative to the shape. Four clues that can be given during the game: (1) Within the shape; (2) Outside the shape; (3) On an edge but not a vertex; or (4) A vertex. The seekers use the clues, their knowledge of a coordinate system, and their knowledge of the special properties of geometric shapes to find the vertices in as few guesses as possible.

Assessment Connections

Questioning . . .

Open with . . .

• How do you read or identify a point on a coordinate plane?

Probe further with . . .

• Which number in the ordered pair represents the x value? y value?
• What information does the first number in each ordered pair tell us?
• If you know the ordered pair, how do you graph the point?
• What information does the second number in each ordered pair tell us?

Listen for . . .

• Can the student identify the ordered pair given a point in the plane?
• Can the student describe how to graph a point?
• How does the student locate a point on a grid given the ordered pair?
• Can the student support the accuracy of the plotted point?
• Does the student use directional words such as "right" and "up" correctly with x and y values?

Look for . . .

• Does the student graph points appropriately?
• Can the student name the ordered pair for a given point?
• Does the student start at the origin to locate an ordered pair?

TAKS Connections

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

• Spring 2004, grade 6, item 1
• Spring 2006, grade 6, items 35 and 39

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(6.8) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles.

(6.8.a) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles. The student is expected to estimate measurements (including circumference) and evaluate reasonableness of results.

Clarifying Activity with Assessment Connections

Students create a poster using estimations of length, area, time, temperature, volume, weight, or angles to show relationships. A student picks an object, for example a bathtub, and estimates its volume using the volume of familiar containers.

Example: A standard bathtub will hold:

• 40 gallons of milk
• 51 three-liter sodas
• 427 cans of soda
• 640 cups of coffee

Example: In the time it takes to watch a movie I could:

• Watch a second hand go around 120 times
• Drive from Corpus Christi to San Antonio
• Watch 4 episodes of a half-hour sitcom
• Listen to 46 songs
• Listen to 1 1/2 CDs
• Boil 12 eggs, one right after another
• Watch 8 periods of basketball
• Attend 2 classes

Assessment Connections

Questioning . . .

Open with . . .

• How would you estimate the measurement of something using standard or nonstandard units of measure?

Probe further with . . .

• What kind of measurements do you need to find?
• How did you determine how many of one object it would take to measure another object?
• Could you use a table to organize your information?
• How do you know your estimates are reasonable?
• How do you estimate your measurement?
• How do you estimate the measurement of an irregular shape?
• How could you find the actual measurement of items?

Listen for . . .

• Is the student using appropriate vocabulary?
• Is the student making reasonable comparisons of measurement?
• Does the student articulate the difference between units of measure?

Look for . . .

• Does the student use an organized method to find the measurements?
• Does the student record the results?
• Are the student's estimates reasonable?
• Does the student differentiate between units of measure?
• Does the student use appropriate units of measure?

TAKS Connections

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

• Spring 2004, grade 6, item 30
• Spring 2006, grade 6, items 24 and 37

(6.8.b) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles. The student is expected to select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), area, time, temperature, volume, and weight.

Clarifying Activity with Assessment Connections

Students select appropriate units, tools, and formulas to measure common items found around them. For example, students may choose a basketball. They decide what attributes of a basketball can be measured (e.g., circumference, diameter, weight), then use appropriate tools and formulas to find these measurements and record them using the correct units of measure.

Assessment Connections

Questioning . . .

Open with . . .

• What attributes of a basketball can be measured and what are the measurements?

Probe further with . . .

• How can you measure these attributes? What tools can you use?
• What can the measurements tell us?
• What possible units could you use when measuring the basketball? Why would you choose these?
• Can you describe at least two ways you could find the diameter of a basketball? (Use a formula based on the circumference or mark on a piece of paper the widest parts and measure between.)
• If you measured the circumference of the basketball, how could you use this measurement to find the basketball's diameter? What formula could you use?

Listen for . . .

• Can the student state the relationship between different measurements?
• Can the student express the measuring process correctly?
• Is the student using the appropriate vocabulary?

Look for . . .

• Is the student's measurement reasonable?
• Is the student recording the measurements appropriately?
• Is the student using appropriate units of measure?

TAKS Connections

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

• Spring 2004, grade 6, items 16 and 32
• Spring 2006, grade 6, item 28

(6.8.c) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles. The student is expected to measure angles.

Clarifying Activity with Assessment Connections

Given a set of different triangles or quadrilaterals, students measure the angles (using a protractor or geometry exploration software) and predict a relationship. Students cut off each of the angles of one of the polygons and piece the angles together with a common point to further explore the angle-sum relationships (e.g., the sum of the angles of the triangle is 180 and the sum of the angles of a quadrilateral is 360).

Assessment Connections

Questioning . . .

Open with . . .

• How do you measure an angle?

Probe further with . . .

• How could you use a table to organize your information?
• What is the sum of the angles you measured?
• After you have cut off and pieced the angles together with a common vertex, what do you notice about the arrangement?

Listen for . . .

• Is the student using appropriate geometric language to describe the angle and the measurement of the angle?
• Does the student express the relationship between the arranged angles of the triangle and a line?
• Does the student express the relationship between the arranged angles of quadrilaterals and a complete circle?

Look for . . .

• Is the student using a protractor or the geometry software correctly?
• Does the student organize the data he or she collects?
• Does the student recognize that the angles of the triangle form a straight angle?
• Does the student recognize that straight angles measure 180 degrees?
• Does the student recognize that the sum of the angles of a quadrilateral add up to 360 degrees?
• Does the student recognize that the number of degrees in a circle is 360?

TAKS Connections

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

• Spring 2003, grade 6, item 19
• Spring 2004, grade 6, item 7
• Spring 2006, grade 6, item 30

(6.8.d) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles. The student is expected to convert measures within the same measurement system (customary and metric) based on relationships between units.

Clarifying Activity with Assessment Connections

Students create a table to illustrate the relationship between two units of measure within the same system, such as centimeters and meters. For example, by looking at meter sticks or tape measures, students can record the first few items in the table and then make a general statement about the relationship between the units of measure. This may be extended to include several units of measure within the same system.

Assessment Connections

Questioning . . .

Open with . . .

• What relationships exist between two units in a measurement system?

Probe further with . . .

• Which unit is smaller? Larger?
• Do you see patterns in the table you created?
• What are the units of measure in the customary system? Metric system?
• What are some appropriate units of measure for length, weight, capacity, etc.?
• What additional relationships exist within these units of measure?

Listen for . . .

• Can the student verbalize the relationships from the tables?
• Can the student compare the relative size of each unit?
• Is the student using vocabulary (such as greater than, less than, larger, smaller, etc.) correctly?

Look for . . .

• Can the student identify the relationship between the units?
• Can the student represent the relationship between the units symbolically?
• Can the student identify the units of measure?

TAKS Connections

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

• Spring 2003, grade 6, item 9
• Spring 2004, grade 6, item 46
• Spring 2006, grade 6, item 43

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(6.9) Probability and statistics The student uses experimental and theoretical probability to make predictions.

(6.9.a) Probability and statistics The student uses experimental and theoretical probability to make predictions. The student is expected to construct sample spaces using lists and tree diagrams.

Clarifying Activity with Assessment Connections

Students conduct an experiment in which they roll two number cubes—a red one and a green one—and determine all possible outcomes (sample space): (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), etc. The first element of each ordered pair represents the outcome of the red number cube and the second element represents the outcome of the green number cube.

Assessment Connections

Questioning . . .

Open with . . .

• What organizational tool can be used to construct a sample space? What are all the possible outcomes?

Probe further with . . .

• How could you organize the information from this situation in a different way?
• How do you figure out how many possible outcomes there are?
• How many possible outcomes are in your sample space?
• How do you know that you have listed all of the outcomes?
• Would a tree diagram help you in this situation? Why or why not?

Listen for . . .

• Can the student explain how he or she determined the sample space?
• Can the student justify that he or she has found all the possible outcomes?
• Does the student have a systematic way of finding his or her sample space?

Look for . . .

• Is the student's sample space well organized?
• Can the student show all possible outcomes?
• Is the student using an effective strategy to construct his or her sample space?

TAKS Connections

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

• Spring 2003, grade 6, item 22
• Spring 2004, grade 6, item 43
• Spring 2006, grade 6, item 6

(6.9.b) Probability and statistics The student uses experimental and theoretical probability to make predictions. The student is expected to find the probabilities of a simple event and its complement and describe the relationship between the two.

Clarifying Activity with Assessment Connections

The teacher assembles three bags of colored cubes or marbles. The bags should be assembled in such a way that the probability of drawing one color (red, for example) will be different for each bag.

Students plan and conduct experiments to determine the probability of drawing a red cube or marble from each bag. For each bag students will also determine the probability of drawing a cube or marble that is not red. After students have made their predictions, they empty the bags and compare their predictions with the contents of the bags.

The teacher can initiate a discussion with students about the relationship between the probability and its complement. During this discussion, the teacher can expect students to observe that the sum of a probability (simple event) and its complement is 1. (Drawing a marble that is not red is the complement of drawing a marble that is red, or, the two sets of outcomes do not have any outcomes in common and include all the possible outcomes in the sample space.)

Students can use this generalization to determine the probability of the complement of a given event. For example, if the probability of spinning a 3 on a spinner is 3/4, then the probability of not spinning a 3 (the complement of spinning a 3) is 1 minus 3/4 = 1/4.

Assessment Connections

Questioning . . .

Open with . . .

• For each bag, how could you predict the probability of drawing a red and the probability of not drawing a red?

Probe further with . . .

• How did you decide when you had completed enough trials?
• What is your prediction for how many of each color do you think are in each bag?
• What is your prediction for what part of the marbles/cubes you think are red?
• What is your prediction for what part of the marbles/cubes you think are not red?
• After you have emptied the bags, how do you determine the probability of drawing a red marble? Of drawing a marble that is not red?
• What is the relationship between the simple event and its complement?
• What do you notice about the relationship between the probabilities of drawing a red and not drawing a red for each bag?

Listen for . . .

• Does the student use appropriate vocabulary to describe probability?
• Can the student reason that the sum of the probability of a simple event and the probability of its complement is equal to 1 and justify this conclusion?

Look for . . .

• Is the student able to make predictions based on their experimental probability?
• Is the student able to find the probability of an event and its complement?
• Does the student notice that the sum of the probability of an event and the probability of its complement are equal to 1?
• What strategies does the student use to determine the complement (subtraction, counting, etc.)?
• Is the student writing the probability as a fraction? As a percentage?

TAKS Connections

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

• Spring 2003, grade 6, item 25
• Spring 2004, grade 6, item 35
• Spring 2006, grade 6, item 23

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(6.10) Probability and statistics. The student uses statistical representations to analyze data.

(6.10.a) Probability and statistics. The student uses statistical representations to analyze data. The student is expected to select and use an appropriate representation for presenting and displaying different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot.

Clarifying Activity with Assessment Connections

The teacher posts a series of survey questions on the walls of the classroom. Each question should have four or more possible responses. Students do a gallery walk and respond to each survey question. Students tally the results and create frequency tables.

Using one of these frequency tables, students will construct appropriate graphs. They may construct a line plot, line graph, bar graph, or stem-and-leaf plot. Each student will construct at least one graph; however, the objective here is for each set of data to be represented in a variety of ways.

Assessment Connections

Questioning . . .

Open with . . .

• How are you going to graph the results of our surveys?

Probe further with . . .

• How will you label your graph?
• What information should your graph show?
• How is the data represented in your graph?
• How is this method of displaying your data different from the other methods?
• Could you represent this data in a bar graph? A circle graph? A line plot? A stem-and-leaf plot?

Listen for . . .

• Is the student able to identify the features of his or her graph?
• Is the student able to justify the representation he or she has selected?
• Is the student able to compare the different representations?

Look for . . .

• Has the student selected an appropriate representation?
• Has the student labeled the graph appropriately?
• Has all the data been included correctly?
• Is the data displayed in more than one type of graph?

TAKS Connections

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

• Spring 2003, grade 6, item 1
• Spring 2004, grade 6, item 4

Extension Using Technology

After students have constructed their own graphs, they can use a calculator or computer software to make additional comparisons with this data or other sets of data. (StatCrunch, at www.statcrunch.com, provides data analysis tools students can use to create graphical representations.)

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

(6.10.b) Probability and statistics. The student uses statistical representations to analyze data. The student is expected to identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data.

Clarifying Activity with Assessment Connections

Students use concrete objects and pictorial models to investigate and identify the mean of a data set. For example, students are told that when asked about how many pets they have, seven families reported the following numbers: 1, 3, 3, 3, 5, 6, and 14. Students represent the number of pets in each family with towers of Unifix cubes as shown below.

Ask students how they might redistribute the pets so that each family has the same number. Students disassemble the cubes and reassemble them into towers representing the same number of pets per family. The students discuss results and the processes that they used. [Note: Although students are not required to use an algorithm to find mean until grade 7, they may have had experience using a calculator to find mean, or they may share that they already know how to find mean. If so, you can ask them to relate their process to the algorithm for finding the mean of the seven data points for which one adds the number of pets and divides by the number of families.]

Students next investigate other data sets resulting from 35 cubes split over 7 towers. Their investigation leads them to the conclusion that each data set has a mean of 5.

These collections represent results that could have come from asking 7 families about the number of pets they own. Students discuss the number of pets as it relates to the towers.

For example, one data set with a mean of 5 is

This collection of towers would represent seven families reporting 1, 5, 5, 5, 6, 6, and 7 as the number of pets they have. When guided, students create a line plot like the one below to show how many families have a specific number of pets.

Still another example of a collection with mean 5 could represent seven families reporting 4, 5, 5, 5, 5, 5 and 6 as the number of pets they have, as shown below.

Students create a line plot like the one below to show how many families have a specific number of pets for this data.

Students discuss which of the two examples of collections with mean 5 is closer to having an equal distribution. A way for students to start to consider which is closer to having an equal distribution is to count how many towers vary from the mean. In this context, how many families differ from the mean pet number of 5? Two families differ from the mean in the last example. Four families differ from the mean in the previous example.

Students then use the line plots to examine the idea of mean pictorially as a balance point. Students revisit the line plots and replace the Xs with numbers indicating how far above or below the mean the X is. For example 1, since the mean was 5, the tower with one cube was 4 cubes short, the towers with 6 cubes were 1 cube over, and the tower with 7 cubes was 2 cubes over.

For example 2,

The students notice that the sum of values below the mean is the same as the sum of the values above the mean in this embellished line plot. This will always be the case. Students note that for this reason, it is said that the distribution of the data will "balance" at the mean and the mean is thought of as the balance point of the distribution.

Assessment Connections

Questioning . . .

Open with . . .

• How might we use concrete objects and pictorial models to examine the typical number of pets for these seven families?

Probe further with . . .

• Notice that the number of pets a family owns varies. What if we used all of the pets and tried to make all families have the same number, in which case there is no variability? How many pets would each family have?
• What is the mean number of pets a family owns? How do you know?
• In words, explain the algorithm that you can use to find the mean of a data set.
• (Optional for students who have prior experience finding mean with a calculator.) How does the process you used, combining all of the pets and then splitting them evenly among the families, relate to the algorithm for finding the mean?
• What if we know that the mean number of pets for seven families is 5? What are some different cube representations that might produce a mean of 5?
• Describe the number of pets each family had as represented by this collection of cube towers.
• In which collection does the number of pets vary more from the mean?
• How many pets does each family have based on the cube towers? How many families have no pets? How many families have one pet? Two? Three? Four? Five?
• Can you create a line plot to show how many families have a specific number of pets?
• What do you notice about the sum of the values below the mean and the sum of the values above the mean in this embellished line plot?
• Why might the mean be thought of as the balance point of the distribution?

Listen for . . .

• Does the student understand the notion of equal distribution for a set of numeric data?
• (Optional for students who have prior experience finding mean with a calculator.) Can the student apply the algorithm for finding the mean?
• Does the student recognize the relationship between modeling the action with concrete objects and applying the formula to find the mean?
• Can the student describe the number of pets each family has as represented by the cube towers?

Look for . . .

• Can the student create a line plot to show how many families have a specific number of pets?
• Does the student notice that the sum of the values below the mean is the same as the sum of the values above the mean in the embellished line plot?
• Can the student identify the mean given towers of cubes that represent the data?
• Can the students identify the mean given a line plot of the data?

TAKS Connections

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

• Spring 2003, grade 6, item 26
• Spring 2004, grade 6, item 13
• Spring 2006, grade 6, item 10

Extension Using Technology

Students can input data from this activity in their calculators or computer and then change one of the values and determine the effect of the change.

(6.10.c) Probability and statistics. The student uses statistical representations to analyze data. The student is expected to sketch circle graphs to display data.

Clarifying Activity with Assessment Connections

Students sort themselves by the month of their birth and form a human bar graph. The last person in January joins hands with the first person in February and so on, so that the students in the bars join to form a circle. The teacher stands at the center of the circle and uses string or adding machine tape to mark the sectors of the circle from the center to the point between each of the groups of students designating the different months. Students estimate the fraction of the circle represented by each month and sketch the graph that results.

Assessment Connections

Questioning . . .

Open with . . .

• What is the relationship between the bar graph and the circle graph of the data?

Probe further with . . .

• What information is available in the bar graph that is not available in the circle graph?
• What information is available in the circle graph that is not available in the bar graph?
• Can you make a circle chart from the bar graph?
• Can you estimate the percentages using equivalent fractions?

Listen for . . .

• Is the student able to identify the important features of either representation?
• Does the student use mathematical terminology to describe the graphs?
• Does the student make comparisons between the frequencies of the data to describe the graphs?

Look for . . .

• Is the student able to represent the fractions correctly on the circle graph?
• Can the student draw a circle graph from the data that was collected and represented?
• Does the student label the graphs appropriately?

TAKS Connections

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

• Spring 2003, grade 6, item 46
• Spring 2004, grade 6, item 22
• Spring 2006, grade 6, item 1

(6.10.d) Probability and statistics. The student uses statistical representations to analyze data. The student is expected to solve problems by collecting, organizing, displaying, and interpreting data.

Clarifying Activity with Assessment Connections

The teacher gives students the following scenario:

You have been asked to help the cafeteria plan its menu. You will need to design and conduct a schoolwide survey to determine the favorite foods served in the cafeteria, and then analyze the data. Represent the data in an appropriate graph, draw conclusions from the data, and use the data to present a suggested menu to the cafeteria manager.

Note: Some schools require school board approval before students conduct a schoolwide survey.

Assessment Connections

Questioning . . .

Open with . . .

• How would you conduct a survey and analyze the data?

Probe further with . . .

• What questions will you ask on your survey? Why did you choose those questions?
• How will you conduct your survey?
• How will you organize the results from the survey?
• How will you evaluate your results?
• How will you use the results of your survey?
• How will you represent your data?
• Can you report your results as an estimated percentage of the total surveyed?

Listen for . . .

• Is the student asking questions that produce categorical data?
• Does the student know the difference between numerical and categorical data?
• Is the student correctly analyzing and interpreting the data?
• Is the student drawing conclusions that are representative of the data collected?

Look for . . .

• How does the student collect the data?
• Can the student organize his or her data?
• Is the student displaying data correctly?
• Is the student using appropriate labeling?
• Is the student's interpretation of the data reasonable?

TAKS Connections

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

• Spring 2003, grade 6, item 6
• Spring 2004, grade 6, item 20
• Spring 2006, grade 6, item 20

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(6.11) Underlying processes and mathematical tools. The student applies Grade 6 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

(6.11.a) Underlying processes and mathematical tools. The student applies Grade 6 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 6.3C, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 6.2C, 6.2D, 6.8A, 6.8B, 6.10B, 6.10D.

TAKS Connections

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

• Spring 2003, grade 6, items 3 and 33
• Spring 2004, grade 6, item 36
• Spring 2006, grade 6, items 8 and 33

(6.11.b) Underlying processes and mathematical tools. The student applies Grade 6 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 6.10D, Probability and statistics. Additional activities that exemplify this student expectation include 6.2B, 6.2C, 6.2D, 6.3C, 6.8B, 6.9B.

TAKS Connections

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

• Spring 2004, grade 6, items 18 and 33
• Spring 2006, grade 6, item 5

(6.11.c) Underlying processes and mathematical tools. The student applies Grade 6 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 6.2A, Number, operation, and quantitative reasoning. Additional activities that exemplify this student expectation include 6.2B, 6.4A, 6.4B, 6.6B, 6.9B, 6.10A.

TAKS Connections

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

• Spring 2004, grade 6, items 8 and 24
• Spring 2006, grade 6, item 12

(6.11.d) Underlying processes and mathematical tools. The student applies Grade 6 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 6.8B, Measurement. Additional activities that exemplify this student expectation include 6.2A, 6.8A, 6.9B, 6.10D.

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(6.12) Underlying processes and mathematical tools. The student communicates about Grade 6 mathematics through informal and mathematical language, representations, and models.

(6.12.a) Underlying processes and mathematical tools. The student communicates about Grade 6 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 6.5A, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 6.2D, 6.3B, 6.4B, 6.6A, 6.8A, 6.8C, 6.10A, 6.10B.

TAKS Connections

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

• Spring 2004, grade 6, item 27
• Spring 2006, grade 6, items 22 and 42

(6.12.b) Underlying processes and mathematical tools. The student communicates about Grade 6 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 6.10A, Probability and statistics. Additional activities that exemplify this student expectation include 6.2B, 6.2D, 6.9A, 6.9B, 6.10B, 6.10C.

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

(6.13.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 6.3A, Patterns, relationships, and algebraic thinking. Additional activities that exemplify this student expectation include 6.1A, 6.1D, 6.1E, 6.4A, 6.4B, 6.8A.

TAKS Connections

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

• Spring 2003, grade 6, item 17
• Spring 2004, grade 6, item 3
• Spring 2006, grade 6, item 46

(6.13.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 6.6C, Geometry and spatial reasoning. Additional activities that exemplify this student expectation include 6.2B, 6.4B, 6.8B, 6.8C, 6.9B, 6.10B.

TAKS Connections

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

• Spring 2003, grade 6, item 8
• Spring 2006, grade 6, item 40

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