Math7Sample.htm

Sample Tasks for PreK-8, developed by New York State teachers, are clarifications, further explaining the language and intent of the associated Performance Indicators. These tasks are not test items, nor are they meant for students' use.

Strands

Process

Content

Number Sense and Operations

Algebra

Geometry

Measurement

Statistics and Probability

Problem Solving Strand

7.PS.1 Use a variety of strategies to understand new mathematical content and to develop more efficient methods

Divide the class into groups of four. Give each group a pile of manipulatives that are not a multiple of four. Have students record the total number of manipulatives in their group.

Round 1: Each person in the group gets one manipulative. On paper students show the original number of manipulatives less the number distributed.

Round 2: Students continue to distributive manipulatives and record each subtraction until there are not enough manipulatives for a complete round.

Relate the results to division, remainders, decimal and fraction representations. From this activity the students should develop a more efficient method to do this task, using division. Have students use the calculator to obtain the number of items each person will eventually obtain.

Students should be able to mathematically obtain the remainder with the calculator.

According to the Humane Society, a cat and her offspring produce an average of 420,000 kittens in 7 years. Write the number of kittens produced by a cat and her offspring in 7 years as a number in scientific notation. If this rate continues, write the number of kittens that would be produced after 14 years.

7.PS.3 Understand and demonstrate how written symbols represent mathematical ideas

Expand and rename the fractions below by factoring out factors common to the numerator and denominator.

Ask students to make a generalization about dividing exponents with a common base.

Students will solve problems that arise in mathematics and in other contexts

The information in the table below shows the cost to enter the Fun House at the carnival. Determine how much it will cost for a family of 8 to enter the fun house. How much will it cost for a family size of n?

Number in Family                          Cost of Admission
1                                                         3.50
2                                                         4.00
3                                                         4.50
4                                                         5.00
5                                                         5.50
6                                                         6.00

List the numbers 7, 9, 13, 16, and 20. Ask students to identify which numbers are perfect squares and which numbers are non-perfect squares. Find the value of the non-perfect squares to the nearest thousandths.

7.PS.6 Represent problem situations verbally, numerically, algebraically, and graphically

A cellular phone company is offering a new phone service exclusively for students. They offer a monthly plan for $9.99 per month, including unlimited local and long distance phone calls and $0.25 per text message.

When would a student's cellular phone bill be more than $20 per month under this new cellular phone service plan? Describe how you would solve this problem and then show the work.

Students will apply and adapt a variety of appropriate strategies to solve problems.

7.PS.7 Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages

7.PS.8 Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem

Ask students to find the sum of the numbers 1 to 1,000. Have them create a simpler but related problem. Did they notice a pattern? Share solutions.

The first class bell at Joleen's school rings at 8:05 am. It takes her 13 minutes to walk to school. Joleen wants to get to school early so that she can visit with her friends for 10 minutes before going to homeroom. It takes Joleen 65 minutes to get ready for school. When the alarm goes off in the morning, Joleen always presses the snooze bar on her alarm clock so that she can get an additional 15 minutes of sleep. What time should Joleen set her alarm so that she will be on time for her first class?

The students from Mountain View School are taking a 154-mile trip to the Catskill Mountain Region. After 45 minutes on the bus, the students asked the bus driver, "How many more minutes until we will arrive?" The bus driver said, "We have gone 33 miles so far. If we continue to travel at the same rate, how many more minutes until we arrive?"

Form groups of 4 and distribute one clue to each student in the group. The students may read their clues as many times as needed to the members of their group, but the students may not show their clue to anyone in their group. Have them determine the answer, based on the clues. Find the number.

CLUES:
Student 1: The number is a perfect square.
Student 2: The number is a multiple of 5.
Student 3: The number has exactly 9 factors.
Student 4: The number is an integer.

Students will monitor and reflect on the process of mathematical problem solving

Lakes Junior High School publishes an annual yearbook for the students in grades 7 and 8. Two costs associated with the yearbook are photography for $700 and printing for $2400. The yearbook committee sold advertisements to local businesses for $1600, and they sold 400 yearbooks for $10 each. Have students calculate the net profit for the yearbook committee.

Lee surveyed 120 students and compiled her information in the chart below. She then made a circle graph to represent the information. If she surveyed only students who have a pet, is this graph reasonable? Explain your answer.

Type of Pet	Number of Students Who Own This Type Of Pet
Cats	44
Dogs	32
Reptiles	14
Small Mammals	20
Other	10

Jackie has volunteered to feed her neighbor's cats dinner. She was left with directions that said:

Feed the kittens, Stripes and Patches, the same amount of food.
Amber, the mother cat, gets twice as much food as the kittens.

There is a container with 5 cups of food. Write an equation and determine to how much food each animal should receive.

Find the surface area of a cylinder. Have the students solve the problem using 2 methods. One method is to find the surface area for each face and then add totals together. Method 2 is to apply the formula.

Students will recognize reasoning and proof as fundamental aspects of mathematics.

7.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies

According to the USDA, a 1600 calorie diet should include about 5 ounces of lean meat and beans, and a 2600 calorie diet should include about 7 ounces of lean meat and beans each day. Over a period of 30 days, how many more pounds of lean meat and beans are consumed under the 2600 calorie diet as compared to the 1600 calorie diet? Explain the method used to solve this problem and determine additional methods for solving the problem.

Then have the class determine what happens to the product of two odd numbers if each odd number is increased by 2.

7.RP.3 Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates

Divide the students into pairs. Provide each pair of students with a bag containing an assortment of different colored objects that are the same size. Have students shake the bag and predict what is in the bag (object and color).

Have students remove an object from the bag, note its color, and replace it in the bag. After the students have sampled 10 objects, have them once again predict what is in their bag.

Ask the students to repeat the experiment. Once again have the students predict what is in the bag.

Next, ask the students to remove all the objects from the bag and record the number of objects and the colors of the objects. Have the students sample without replacement until all of the objects are removed from the bag.

Ask students to compare the experimental probability to the theoretical probability of selecting an object.

Rosalina said the product of two factors is always larger than the two factors. She supported her conjecture with examples such as 50 x 2 = 100, 7 x 9 =63.

Sylvan said that multiplication always results in a product larger than or the same as one of the factors. He supported his conjecture with examples such as 3 x 4 = 12, -3 x -4 = 12, 0 x 6 = 0, 1 x 1 = 1. Conduct a discussion of the definition of conjecture. Have the students determine whether Rosalina's or Sylvan's conjecture is accurate.

7.RP.5 Develop, verify, and explain an argument, using appropriate mathematical ideas and language

Review the divisibility rules for 2 and 3. Then have students make a conjecture of the divisibility rule for 6. Ask each student to support their conjecture using mathematical ideas.

Students will select and use various types of reasoning and methods of proof.

7.RP.6 Support an argument by using a systematic approach to test more than one case

Using a protractor, have the students draw polygons with 3, 4, 5, 6 and 7 sides, and use a protractor to measure the angle of each polygon. Sum the angle measures for each polygon and record. Do this for 5 polygons and compile the information. From this trial of 5 polygons, have students develop a rule for determining the sum of the interior angles of polygons.

7.RP.7 Devise ways to verify results or use counterexamples to refute incorrect statements

Using two colored counters to subtract integers, have students explain why subtracting a number is the same as adding its opposite.

Then have students explain why subtracting is the same as adding the opposite when using a number line.

7.RP.8 Apply inductive reasoning in making and supporting mathematical conjectures

Students will organize and consolidate their mathematical thinking through communication.

7.CM.1 Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving

Divide students into pairs and have them take turns solving problems. Have one student solve one of the following problems and explain their results to their partner.

Problem 1: The corner newspaper box accepts any of the following: nickels, dimes, quarters or half dollars. It does not accept pennies. The weekday paper costs $.50. What are all of the possible coin combinations that can be used?

Problem 2: A snail is at the bottom of a 10-inch hole. The snail climbs up 2 inches each day but falls back one inch each night. How many days will it take the snail to crawl out of the hole?

Problem 3: Ellen has two copies of a photograph. One is 3 inches by 5 inches, and the other copy is 4 inches by 6 inches. Are these photographs similar rectangles?

7.CM.2 Provide an organized argument which explains rationale for strategy selection

Have students reach into a box and select a problem to solve. Have students choose the best strategy for solving their problem. The student will not solve the problem, but will explain to the class why they believe they selected the best strategy for solving the problem.

1. Sam's dad gave him three wooden boards to make a bookcase. All the shelves need to be the same size. The boards are 108 inches, 60 inches and 24 inches. Sam's dad said that there can be no pieces left over. What is the largest sized shelf Sam can make?

2. At the end of the year party Lisa brought a big box of doughnuts. The boys ate half the donuts; the girls ate 6 of the donuts. The teachers ate as many as the girls ate and there were 3 left over. How many dozen donuts did Lisa bring?

3. The school photographer is taking pictures for the yearbook. There are 27 pictures on each roll of film. The photographer takes 4 pictures of each student. How many rolls of film are needed to take pictures of 185 students?

4. The math class is planning a party. There are 29 students in the class. 10 students want plain cheese pizza, 15 students want pepperoni pizza and 4 students do not care what kind of pizza they eat. The large pizzas are cut into 8 pieces. What is the fewest number of pizzas that can be ordered so that each student gets two pieces of pizza?

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

7.CM.4 Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form

The Beverage Barn Cola is on sale for 6 cans for $2.22. How much does one can cost? Create a drawing, a chart, and a paragraph that describe this problem and solution.

Divide the class into small groups. Provide each group of students a picture of an item (e.g., elephant, truck, cat, mouse, large marble column, grain of sand, baby, planet). Have each group propose the best tool and technique to measure the mass of their item. Present the solution to the class and ask students to ask clarifying questions and state why they believe the tool and technique were appropriate.

Students will analyze and evaluate the mathematical thinking and strategies of others.

Divide the class into small groups. Provide small groups of students with an empty sealed box in the shape of a rectangular prism, and a single square that has a 2 cm side. Have students determine the surface area of the box in square centimeters and explain their answer to the class.

7.CM.7 Compare strategies used and solutions found by others in relation to their own work

Anna says that the expression a² means the same as a x 2 because if a= 2 then 2²= 4 and 2 x 2 =4. Is Anna correct? Explain your answer.

7.CM.8 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others

Have students discuss how they would solve the problem below and create additional questions on related problems:

Given the set of numbers 7, 14, 21, 28, 35, 42, find a subset of these numbers
that sum to 100.

Students will use the language of mathematics to express mathematical ideas precisely.

7.CM.9 Increase their use of mathematical vocabulary and language when communicating with others

Ask two students to sit back to back. Have one student draw a three-dimensional figure and then direct the other student to draw the same figure using only mathematical terms. When finished, ask the students to compare the original drawing with its copy. Ask each pair of students what their drawing represents. Discuss how mathematical vocabulary was helpful in completing the drawing.

7.CM.10 Use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

Hon's family has a rectangular-shaped pool that is 14 feet by 20 feet. They need to put one-foot square tiles around the pool for a walkway. Determine the dimensions of the entire area that includes the pool and walkway if they only use one row of tiles all the way around the pool. How many square feet will the pool and walkway occupy in total? Draw a picture, labeling it with mathematical terms that show the problem. Explain your answer.

7.CM.11 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing

Given the information below, predict which color one would expect to select from a box containing the same items in white, green, red, and orange. Predict the minimum number of items in the box. Explain your predictions.

7.CN.1 Understand and make connections among multiple representations of the same mathematical idea

Jose received grades of 40, 70, 60, 70, and 90 on 5 math tests. Which is a best measure of his progress in class: mean, median or mode? Explain your answer.

Have students determine how many handshakes will take place in the room if everyone shakes everybody else's hand only once. Have students make a conjecture on the number of handshakes that will take place. Discuss strategies that can include solving a simpler problem (e.g., only 4 students, then 6 students, etc.), drawing a picture, or acting it out.

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

7.CN.4 Model situations mathematically, using representations to draw conclusions and formulate new situations

7.CN.5 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

Students will recognize and apply mathematics in contexts outside of mathematics.

7.CN.6 Recognize and provide examples of the presence of mathematics in their daily lives

The graph below shows the number of students who participated in sports at Milla Middle School. Have the students discuss enrollments in the different sports in 2003 and 2004 and draw conclusions based on the graph.

7.CN.7 Apply mathematical ideas to problem situations that develop outside of mathematics

The Green Valley Middle School is planning a trip to Toronto, Canada. How much United States money will Sally need to purchase a souvenir bear that costs $9.95 Canadian? Use the currency conversion $1.00 US: $1.23 Canadian.

7.CN.8 Investigate the presence of mathematics in careers and areas of interest

Have each student interview adults, asking how they use mathematics in their job. The students can either write their interview, or incorporate it into a career poster to share with the class.

7.CN.9 Recognize and apply mathematics to other disciplines, areas of interest, and societal issues

The chart below includes the population of New York State according to the U.S. Census Bureau for 1960, 1980, and 2000.

Calculate the population increase from 1960 to 1980, and 1980 until 2000. In which 20 year period was the population increase greater? Discuss significant historical events that occurred in New York State in 1960, 1970, and 2000.

Students will create and use representations to organize, record, and communicate mathematical ideas.

7.R.1 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

7.R.2 Explain, describe, and defend mathematical ideas using representations

Select a starting and ending point for a car trip of local interest. Provide each student a map and ask the students to calculate how many miles the trip will be, using the roads on the map. Ask students how they might calculate the distance a different way.

Approximately how long will it take to arrive at their destination in a car if their average speed is 45 miles per hour?

7.R.5 Use standard and non-standard representations with accuracy and detail

Students will select, apply, and translate among mathematical representations to solve problems.

Using cubes, have students show and sketch all of the different rectangular prisms that can be made using 24 cubes. Then use the volume formula to confirm the results. Ask the students if they can make a cube using the 24 cubes.

Based on the information in the table below, which type of disk can store the most amount of information? Explain your answer. State the range of the data.

7.R.8 Use representation as a tool for exploring and understanding mathematical ideas

Have students draw a right triangle on grid paper with the legs of the triangle as an integral number. From another piece of grid paper, have students draw and cut out 3 squares, as follows: one square has sides that are the same length of one leg of the triangle, the second square has sides that are the same length of the second leg of the triangle, and the third square has sides that are the length of the hypotenuse of the triangle. Have students verify that the sides of the squares each match up one of the sides of the triangle. Have students cut up the two smaller squares and rearrange the pieces so that the larger, third square is completely covered. Have students state the relationship between the lengths of the three sides of the right triangle.

Students will use representations to model and interpret physical, social, and mathematical phenomena.

7.R.9 Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects)

In technology class, Garret is making a scale drawing of a CD case. In his drawing the CD case is 2 inches high. When he makes his CD holder it will be 2 feet 6 inches. What is the scale of his drawing?

7.R.10 Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks)

The seventh grade class is planning a dance. The disc jockey will cost $250 for 4 hours and it will cost $100 to decorate the gym. If there are 270 students in the school and two-thirds of the students have purchased tickets for $5, complete a table of values and construct a graph to show how much profit the seventh grade will make.

7.R.11 Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function)

Provide measuring tapes, calipers, rulers and a variety of cans, lids, and other cylinders or circular objects. Have students measure a variety of cylindrical objects and record the diameter and the circumference. Students may have to trace the circumference on paper to measure the diameter, or use calipers. Have students create a table and ask them to determine the relationship between the diameter and circumference.

Number Sense and Operations Strand

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.

7.N.1 Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers, integers, rational numbers, and irrational numbers)

Draw a concept map to display the relationship among natural/counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. Place numbers from each of the subsets on index cards. Using tape or string place a large concept map on the floor and give each student an index card. Have the students arrange themselves in the concept map according to the subset in which the number belongs. Below is an example of a concept map.

7.N.2 Recognize the difference between rational and irrational numbers (i.e., explore different approximations of )

Place various rational and irrational numbers on the board in two separate lists and have the students, using calculators, make observations and create additional examples. Have students explain why each number is placed in List A or List B.

Ask students to explain the difference between  and its various approximations: 22/7, 3.14, 3.14159, etc. and have them place them on a number line.

7.N.3 Place rational and irrational numbers (approximations) on a number line and justify the placement of the numbers.

Place a variety of rational and irrational numbers on index cards and distribute the cards to students. Have students fasten their cards to a clothesline stretched across the room. As numbers are added to the line, students may have to adjust the spacing of some of the cards already placed on the "number" line. Hang a final card that has a question mark on it. Have the class guess what number it represents.

Make up several examples, such as the following, and have the students write them in standard form using what they already know about exponents.

Ask the students for their observations of the factors, products, bases, and exponents. Have them develop the law of exponents for multiplication. Do a similar process for division based on what students already know about exponents and factoring forms of one, and after several examples, have them make observations to guide them to the law of exponents for division.

Continue to have the students factor out forms of one to arrive at the answer in simplest form. Continue with several more examples such as the following:

Make up several decks of cards for each group of students to play Challenge, the card game that students may know as War. Each card, written in standard form, should represent one large number, billions or higher, or one small number, millionths or smaller. Some of the cards should have the same value. To play, deal all the cards to a small group of players. All players lay down the top card from their hand and highest card wins the pile. In case of a tie, challenge is declared and those players lay two cards face down and one face up to decide who wins the pile. Again, high card wins. Play this for a few minutes to give the students the idea of how difficult it is to read these numbers.

Then supply the students with blank cards and direct each group to convert their deck of standard notation numbers into scientific notation.

Play Challenge as described in 7.N.5a, but use the cards that the students converted into scientific notation.

7.N.8 Find the common factors and greatest common factor of two or more numbers

Venn diagrams can be used in a variety of ways to show common factors and greatest common factors of two or more numbers. For example:

a) Using Venn diagrams, give students two or three numbers and have them place the factors of those numbers in the appropriate sections.

b) Place factors of numbers, one at a time, in the sections of a Venn diagram. As you are adding factors to the diagram, have students try to guess the rule. Then have them label the Venn diagram.

c) Have students make up partial Venn diagrams and give them to partners to complete.

At a party, the prize box contained enough one-dollar bills so that one to six winners could share it equally. What is the least amount of money that could be in the prize box?

7.N.10 Determine the prime factorization of a given number and write in exponential form

a. x3²x5 is the prime factorization of 90.
b. 3x4x7 is the prime factorization of 84.
c. 2³x3x5 is the prime factorization of 90.

Students will understand meanings of operations and procedures, and how they relate to one another.

7.N.11 Simplify expressions using order of operations Note: Expressions may include absolute value and/or integral exponents greater than 0

Give the students both a calculator that has not been programmed to do order of operations (usually a 4-function calculator) and a scientific calculator that has been programmed to do order of operations. Have students do the operations for the problems below and compare their results.

Give them such problems as 5 + 3 ^. 8 to evaluate. They will get 64 and 29 on the two different calculators. You may want to give them a few more such as 3 + 12 ÷ 4; 17 - 2(2 + 3). Discuss why it is necessary to agree on the order of calculations.

Evaluate the expressions below using order of operations and check using a scientific calculator.

Provide numeric expressions and ask students to relate them to football, money, temperature, positive/negative chips, etc. Then ask them to solve the expression.

Develop a set of cards with word problems and another set of cards that contain the corresponding equations to be used as a matching game. This could be played by the entire class or by groups of student. Make sets of cards for each group. See examples below:

a) How much is Shantel's total worth if she borrowed three dollars each from eight people? 8 x -3 = -24

b) The Patriots lost eight yards on their first play and lost three more yards on the next play. What was their net result after these two plays? -8 + -3 = -11

c) The temperature was 8^o below zero in the morning, and then it rose 3^o. What is the temperature? -8 + 3 = -5

d) Jon's bank statement revealed that he has eight dollars. The bank charged three dollars for checks, but he has free checking so the bank made a mistake. Now they have to take away the three-dollar check charge. What is his balance now? 8 - (-3) = 11.

7.N.13 Add and subtract two integers (with and without the use of a number line)

Using two-color counters where the yellow side represents positive charges and the red side represents negative, have students model and record their results. For example:

Assist the students in explaining each of these problems using such examples as football gain and loss, money, temperature, etc.

Distribute number lines and have students solve a variety of addition problems using the number line, recording their results. Follow this with a variety of subtraction problems using the number line, recording results.

Ask them to explain each as a word problem (e.g., eight points were deducted on a quiz, but the teacher made a mistake and had to take away five of the points that were already taken off on a problem. This results in a loss of only three points.) Then ask students to solve the related addition problems.

Ask students to compare the addition and subtraction problems and discuss how they are related.

Ask students to write a note to a student who is absent explaining the difference between subtracting a negative amount versus subtracting a positive amount and how subtraction and addition are related.

7.N.14 Develop a conceptual understanding of negative and zero exponents with a base of ten and relate to fractions and decimals (i.e., 10^-2 = .01 = 1/100)

List various powers of tens on the board and have students write them out expressed both as factors and their products in standard form.

Have the students describe their observations about the pattern of the exponents, the factors, and the products.

Discuss the pattern of dividing by the base, 10, for the next power of ten. For example: to go from 1,000,000 to the next product of 100,000, you divide by 10. To go from 100,000 to 10,000, you also divide by 10. Have the students discuss this pattern for the remainder of the products.

Following the pattern of subtracting 1 to get the next exponent, it would follow that 10^o would be the next. Using this pattern of dividing by the base, since 10¹=10, the pattern would indicate that 10^o would have to be 10 ÷ 10 = 1. From the pattern of dividing by 10, the understanding of negative exponents can then be developed. Have the students express them both as fractions and as decimals.

7.N.15 Recognize and state the value of the square root of a perfect square (up to 225)

Disk Type	Data Capacity (bytes)
8 inch floppy disk	5 x 10⁵
5.25 inch disk	1.2 x 10⁶
3.5 inch disk	1.44 x 10⁶
CD-Rom disk	6.5 x 10⁸
DVD	8.5 x 10⁹

List A	List B
__ .36	¯
-6
1/2
	
5	3.1011011101111...
3.14
	-.2020020002...
	5
16/3	5

1960	1980	2000
16,782,304	17,558,072	18,876,457