% ! 3 40% % Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents"

Transcription

1 Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A %! Simple fraction! 1. Complete these sets of cards. Decimal! Percent fraction! Percent Set B %! Set C 0.65!! 65% Set D! 3 8!! Copyright 2008 by Mathematics Assessment Resource Service 47

2 2. Show 2, 65% and on the number line below. 0!! 100% 3 4! 7 Copyright 2008 by Mathematics Assessment Resource Service 48

3 Percent Cards Rubric The core elements of performance required by this task are: relate fractions, decimals and percents Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answers: Set B 0.40 Set C 13/20, 65/100 Set D 0.375, 37.5/100 (accept 375/1000), 37.5% Partial credit One error 2. Values correctly indicated on number line. points 1 2 x 1 2 (1) section points % Partial credit One error % 3 4 (1) 2 Total Points 7 2 Copyright 2008 by Mathematics Assessment Resource Service 49

4 Percent Cards Work the task and look at the rubric. What are the big mathematical ideas being assessed in this task? What strategies might you expect students to use to find the simple fraction in Set C? Look at student work on C. How many students gave a correct answer of 13/20? Make a list of other answers. How might students have gotten these answers? Look at student work for set D. How many of your students put: /100 or 375/ %.38 38/100 38% /100 37% /100 30% /100 26% Other What types of strategies did successful students use? What misconceptions did you see as you looked at student work? Did your students show what they were thinking by writing calculations? What does it mean for something to be all the same value? Do students get opportunities to have discussions about when to round and when rounding is not appropriate? If the instructions had told students not to round off, what percentage of the students who did round do you think would be able to make sense of the number in the thousands place and make a correct representation for it? What evidence did you consider in their work? 50

5 Now look at the work on the number line. How many of your students: Could correctly place all the numbers on the number line? Added values on the line to show the scale? Redid the number conversions? How many of your students had values for 3/8 between: 0.375% Other values between 30-39% 20-29% 70-79% 80-89% % Other How many of your students had a relative order of: o 2/5, 3/8/ 65%? o 2/5, 65%, 3/8? Have your students worked with the number line this year? How can this mathematical representation be used to help students solve problems or check their work? How can this representation be used to show the meaning of operations with rational numbers? Have you ever used the double number line to help make sense of percents? What are some other uses of the number line that help students develop a deeper understanding of number? 51

6 Looking at Student Work on Percent Cards Student A uses division to convert the simple fractions to decimals. The student reduces the percent fraction in C to find the simple fraction. The student is able to place all the values on the number line. Student A 52

7 Student B uses a unit fraction approach to thinking about the conversions. If one fifth equals 2/10ths, then 2/5 equals 4/10ths. If 1/20 equals 5/100ths, then 13/20ths equals 65/100ths. Student C makes a common error by reversing the numbers in the division. The student divides the top number into the bottom number. Notice that the student changes the 3 to a 30 so that the answer will be a decimal. How do we help students master a procedure but also learn enough understanding of why it works to help them correct errors as they occur? What kind of experiences could push students thinking about the whys of the procedure? 53

8 Look at the work of Student D. The student seems to know that is equal to 2/3. Can you figure out what the student was doing for the 3/8? Student E is able to complete the division for the 3/8, but doesn t know how to work with the decimal to convert to a percent fraction or a percent? Beyond procedural knowledge, what does a student need to understand about place value and the number system to make these conversions? 54

9 Student F uses two approaches to the conversion of fractions to decimals. In Set B the student finds equivalent fractions. In Set D the student uses division. The student then shows the rounding to go from three to two decimal places. On the second page of the task the student doesn t necessarily connect the values to those on the previous page and reconverts the fractions to decimals and percents. The student can place the fractions correctly on the number line, but misplaces 65%. What might have confused the student? Student F 55

10 Student F, part 2 56

11 Student G adds marks to the number line to scale the values between 30 and 40%. Student G Student H numbers all the lines on the number line to show a scale in percents. The student uses an incorrect value from page 1 of the task for 3/8, but places that value correctly on the line. Student H 57

12 Student I has placed the values in the correct relative order, but does not connect the values to the line. Student I Student J has misplaced the fractions on the number line. The student had correctly identified 2/5=0.40 for Set B and had a value of 3/8 =0.22 for Set B. Notice that neither of these values account for the positions on the number line. What conjecture can you make for the location of the fractions made by the student? Student J Student K has made the common measurement mistake of counting using the 0 as one. All the locations are off exactly 10%. Student K 58

13 6 th Grade Task 4 Percent Cards Student Task Core Idea 1 Number and Operation Relate fractions, decimals and percents. Use number line to locate and compare percents, fractions, and decimals. Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems. Understand fractions, decimals, and percents as parts of unit wholes and as parts of a collection. Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line. The mathematics of this task: Understanding decimal place value and working with values in the thousandths place Converting between representations using fractions, decimals and percents Understanding equivalency Using scale to place values on a number line Based on teacher observations, this is what sixth graders know and are able to do: Convert decimals to percents or percents to decimals Convert decimals to percent fractions and percents Understand the idea of a percent fraction Areas of difficulty for sixth graders: Locating fractions on a number line Reducing 65/100 to a simple fraction Converting fractions to decimals for values smaller than 100ths Understanding equivalency, same value Understanding the relationship between percents on the number line and the value of the fractions Strategies used by successful students: Using calculations from part 1 Marking some other benchmark numbers on the number line, like 1/2, 50%, or 10% 59

14 The maximum score available on this task is 7 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, 94%, could change 2/5 into a decimal and change 0.65 into a percent fraction. Many students, 69%, could also give a simple fraction for 65%. About half the students could find the decimal for 2/5, the fraction and decimal fraction for 65%, and place both these values on the number line. Almost 14% of the students could meet all the demands of the task including changing 3/8 to equivalent decimal and percent values and place 3/8 on the number line. About 3% of the students scored no points on the examine. All the students in the sample with that score did not attempt the task. 60

15 Percent Cards Points Understandings Misunderstandings 0 All the students in the sample with this score did not attempt the task. 2 Students could change 2/5 into a decimal and change 0.65 into a percent fraction. 4 Students could find the decimal for 2/5, the fraction and decimal fraction for 65%, and place both these values on the number line. 7 Students could comfortably convert between simple fractions, decimals, percent fractions and percents. Students could work with values in the 1000ths place. Students could place decimals and fractions on a number line scaled in percents. Students had difficulty changing the 65/100 to a simple fraction. About10% of the students put 2/5 between 20 and 30% on the number line. About 7.5% put 2/5 between 40 and 50%. About 7% put 2/5 between 10 and 20%. Students struggled with 3/8. Almost 9% did not finish the division or rounded to 0.37%. About 7.5% rounded up to 38%. 5% of the students divided 3 into 8 to get 26%. 5% were able to give the decimal of 0.375, but then used 37/100 for the percent fraction. Students also struggled with how to place 3/8 on the number line. Almost 18% had a relative order of 2/5, 65%, 3/8. 9% had a relative order of 2/5, 3/8, 65%. 13% of the students placed 3/8 between 80 and 90%. 10% of the students placed 3/8 between 20 and 30%. 61

16 Implications for Instruction Students need practice relating fractions, decimals and percents. Students should be comfortable with fractions that convert to more than 2 decimal places. Students often know procedures for finding equivalent fractions, but don t understand that equivalent means the same size. Students should be familiar with using a number line to compare numbers. Number lines help students to understand the meaning of adding and subtracting fractions and show how to compare fractions. Models, such as number lines provide students with access to big mathematical ideas, help to justify why the procedures work, and provide students with tools to check their work to see if answers are reasonableness. Models also help students to solve problems and mimic the action of the problems to determine operations. Students at this grade level should be fluent in converting between representations of numbers. Ideas for Action Research Understanding Equivalency, Lesson Study Adults have trouble understanding why concepts are difficult for students. They can t think what is complicated about an idea that they understand. Looking at students expressing their ideas helps to clarify where student thinking breaks down. At an MSRI (Mathematics Science Research Institute in Berkeley) Deborah Ball interviewed a 6 th grade student on equivalent fractions in front of an audience of over 100 math educators from across the U.S. The student seemed confident and willing to talk. The student knew several procedures for working with fractions, but had difficulty explaining why the procedures worked. When probed about placing numbers on the number line, the student reached disequilibrium when 2 different fractions appeared to be located in the same place on the number line. The video can be viewed on line by going to the MSRI website: Then scroll to Communications and click on: streaming video lectures. On the new screen, scroll to Special Productions and click Critical Issues in Mathematics. Finally scroll down to Assessing Student s Mathematical Learning, March This is a good video to view with colleagues. You might discuss the following: What does the student understand? Is this procedural knowledge or conceptual knowledge? Where does the student s thinking breakdown? Why is understanding equivalency so difficult for students? How can we make some of these ideas more explicit when we design lessons on fractions? After viewing the video, your department might consider developing a series of lessons using number lines to make sense of different topics in your curriculum. 62

17 Area and Perimeter This problem gives you the chance to: work with area and perimeter of rectangles 1. The perimeter of this rectangle is 2(5 + 2) = 14 inches. The area of this rectangle is 2 x 5 = 10 square inches. 2 inches a. Draw a diagram of a rectangle with the same perimeter, but a larger area. Write down the area of your rectangle. 5 inches b. Draw a diagram of a rectangle with the same perimeter, but a smaller area. Write down the area of your rectangle. 2. The perimeter of this rectangle is 22 inches. The area of this rectangle is 24 square inches a. Is it possible to draw a rectangle with the same area as the one on the right, but a larger perimeter? Explain your reasoning. 3 inches 8 inches b. Is it possible to draw a rectangle with the same area, but a smaller perimeter? Explain your reasoning. Copyright 2008 by Mathematics Assessment Resource Service 10 63

18 Area and Perimeter Rubric The core elements of performance required by this task are: work with area and perimeter of rectangles Based on these, credit for specific aspects of performance should be assigned as follows point s sectio n points 1.a. Draws a rectangle with sides such as: 3 inches x 4 inches area = 12 square inches 1 1 b. Draws a rectangle with sides such as: 1 inch x 6 inches area = 6 square inches a. Gives correct answer: Yes and Gives correct explanation such as: Area 2 x 12 = 24, Perimeter 2(2 + 12) = 28 inches or Area 1 x 24 = 24, Perimeter 2(1 + 24) = 50 inches 3 or 3 Partial credit Allow partial credit for a partially correct answer. (2) b. Gives correct answer: Yes and Gives correct explanation such as: Area 4 x 6 = 24, Perimeter 2(4 + 6) = 20 inches 3 Partial credit Allow partial credit for a partially correct answer. (2) 6 Total Points 10 Copyright 2008 by Mathematics Assessment Resource Service 64

19 Percents Work the task and look at the rubric. What does a student need to understand to be successful on this task? Look at student work for part 1. How many students gave answers, such as: A complete response with rectangles of correct dimensions that fit the constraints of the same perimeter and a larger or smaller perimeter and quantifified the area for the new shape? Gave rectangles with correct dimensions but didn t give a value for area? Gave rectangles that had larger or smaller areas, but also changed the perimeter? Used the same rectangle of 2 x 5? Gave shapes that were not rectangles? What misconceptions did you see in student work? Now look at the work for part 2a and 2b. How many students gave answers, such as: Used correct dimensions, showed that area is the same, and calculated the new perimeters? Used correct dimensions but didn t quantify why or how they fit the constraints? Thought one was possible and the other was not possible? Thought changing the area would change the dimensions? What other types of errors did you see in students thinking? How do you communicate values around justifying an answer in your classroom? Do students know that quantifying is an important mathematical value? How often do students have the opportunity to do investigations in your classroom? What misunderstandings or habits of mind prevented students from making attempts to find solutions in part 2? Although incomplete, what pieces of thinking showed that students had some understanding of the concepts? What would need to be done to these answers to get a complete solution? How could you use this idea to develop a class discussion? 64

20 Looking at Student Work on Area and Perimeter Student A is able to find rectangles to meet the constraints. The student is able to put numbers into a formula to calculate perimeter and also calculates the area. Notice that the student uses greater than or less than to compare new calculations to originals to complete the proof. Student A 65

21 Student B is able to meet the demands of the task for part 1. In part 2 the student shows an understanding of how to change the shape and restrictions about what types of numbers can be used for the dimensions, but doesn t use specific examples to complete the argument that it is possible to change perimeter without changing the area. Student B 66

22 Student C is able to find rectangles to justify part 2. However the student does not notice the constraint of keeping the perimeter the same, when solving part 1. How do we help students develop the habit of mind to look for all the constraints and check that they have been met? Student C 67

23 Student D is able to find correct dimensions for rectangles to meet the criteria for part 1. In part 2 the student uses different logic for part a and b. In part a the student has the common misconception that a larger perimeter must have a larger area. In part b the student knows that the dimensions need to be factors of 24, but only tests one case which doesn t work. What might be next steps for this student? Student D 68

24 Student E makes nonrectangular shapes in part 1, but still uses the formula for area of a rectangle. How do students develop the connection between shape and possible dimensions? What types of experiences does this student need? In part 2 the student can do part 2b, but doesn t think the dimensions can be changed in 2a. Why do you think students have these kinds of inconsistencies? Student E 69

25 6 th Grade Task 5 Area and Perimeter Student Task Core Idea 4 Geometry and Measurement Work with area and perimeter of rectangles. Investigate how change linear measures effects area and perimeter. Analyze characteristics and properties of two-dimensional geometric shapes. Apply appropriate techniques, tools, and formulas to determine measurements. Develop, understand, and use formulas to determine area of quadrilaterals. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes. The mathematics of this task: Willingness to investigate different cases to see if something is possible Identifying constraints See relationships between dimensions and the measurements of area and perimeter Quantify measurements in order to make comparisons Understanding which dimensional measurements will and will not make a rectangle and why Understanding that rectangles have two sets of parallel sides with the same side length for each set Calculating area and perimeter Understanding the logic of justification Understanding and tracking units Based on teacher observations, this is what sixth graders knew and were able to do: Rectangles have four sides Know formulas for area and perimeter and how to use them to calculate Draw rectangles and give appropriate dimensions Areas of difficulty for sixth graders: Confusing perimeter with diameter, so they kept one dimension the same and changed to height to get the new areas Conducting an investigation. Some students did not believe it was possible to keep the perimeter the same and change the area, but they did not try numbers to test their conjectures. Applying knowledge to a complex problem Understanding that opposite sides of a rectangle needed to be the same size Making connections between area and factors to develop possible side lengths Using all the constraints in the problem Changing the wrong measurement Thought more square-like shapes were no longer rectangles Thought that making changes in one of the dimensions would change both measurements Strategy used by successful students: Thinking about factors to determine possible dimensions for the rectangle 70

26 The maximum score available for this task is 10 points. The minimum score needed for a level 3 response, meeting standards, is 5 points. About half the students could give the dimensions for rectangles with the same perimeter and larger or smaller areas. Some students, about 39% could also calculate the area of the new rectangles. A few students, 30%, could give the dimensions for rectangles with the same perimeter and larger and smaller areas, calculate the areas for the new rectangles, and give the dimensions for a rectangle with the same area and either a larger or smaller perimeter. Almost 14% of the students could meet all the demands of the task including giving the dimensions for a rectangle with the same area and larger and smaller perimeters and justify the dimensions by giving the new perimeters. Almost half the students scored no points on this task. 81% of the students with this score attempted the task. 71

27 Area and Perimeter Points Understandings Misunderstandings 0 81% of the students with this score attempted the task. Students could not give dimensions for rectangles with the same perimeter. Students gave dimensions for shapes that were not rectangular. Students gave shapes 2 Students could give dimensions for rectangles with the same perimeter but larger and smaller areas. 4 Students could give dimensions for rectangles with the same perimeter but larger and smaller areas and justify their solution by calculating the new areas. 5 Students could give dimensions of new rectangles, for parts 1 and 2, but in general did not calculate the area or perimeter for the new shapes. with dimensions that did not add to 14. Students did not justify their solution by calculating the new areas. Students did not understand how to change the perimeter while maintaining the same area. 14% said that if the area is larger, then the perimeter must be larger. Another 8% said that if the area stays the same, the perimeter will stay the same. 10% thought it was possible, but did not attempt to give the new dimensions or test their conjectures. Students don t understand the logic of justification. Once they have a dimension they are done with the task. 8 Students did not calculate the perimeters in part 2 of the task. 10 Students could investigate hypotheses, such as maintaining perimeter and changing the area, and then calculate the values for area and perimeter of the new shape to verify the hypothesis. 72

28 Implications for Instruction Students at this grade level have been working with area and perimeter since third grade. They know the basic procedures for making the calculations on rectangles. However, many students are still struggling with the conceptual knowledge. They haven t learned the information in a way that allows them to apply it to new or unusual situations. Consider the idea of layers of knowledge. One way of knowing something is physical representation (Using only the grid paper provided, construct as many rectangles as you can with an area of 12 square centimetres.) The next level of understanding is to be able to apply an idea to a real world context. (Fred s flat has five rooms. The total floor area is 60 square meters. Draw a plan of Fred s flat. Label each room, and show the dimensions of all the rooms.) Finally, students should be able to work with mathematical abstractions. (The area of a rectangle is 12 square centimetres. What might be the dimensions?) This idea of assessment by contextual exhaustion is from the work of David Clarke, University of Melbourne. As students at this grade level have been working with the calculations and procedures at earlier grade levels, the challenge at this grade level is to work more on cognitive demand by applying the information in new and novel ways, increasing the level of abstraction, and providing problems with longer reasoning chains. Students need opportunities to investigate the relationships between mathematical ideas, such as area and perimeter. They should have tools for organizing their thinking. Students need to develop persistence for solving problems where the correct procedure is not immediately obvious; in this case working from the answer to the dimensions. Students should also be developing criteria for making a convincing argument, by quantifying information and checking it against the original conjectures. Ideas for Action Research Examining Student Misconceptions and Planning a Re-engagement Lesson Almost 49% of the sixth graders scored 0 on this task. 80% of the students with this score attempted the task, amounting to almost 3000 students. For this reason, it is important to look at their thinking in depth. Students have been working with area and perimeter of rectangles since third grade. What made this difficult for students? There are some important misconceptions. Errors are more than just an unwillingness to investigate. The following student work is from students with total scores of 14 to 25. Questions for consideration when examining student work: Are students making the areas larger and smaller in part 1? Are students forgetting to keep the perimeter the same? Are students giving dimensions that will make a rectangle? What does in mean to have a dimension of 0? Is the student changing the wrong measurement, area or perimeter? Is the thinking consistent in both parts of 2, thinking they are both possible or both impossible? What are some of the reasons for the discrepancies? Some students describe shapes that would meet the new criteria for either 2a or 2b, but don t give dimensions or offer proof by verifying area and perimeter. Can you find these? 73

29 Several students use the same dimensions in part one as the sample rectangle. Why might they do this? What aren t they grasping about the prompt? Some of the student reasoning is based on beliefs about the effects of operations on numbers. What is correct and incorrect in their understanding? After examining and discussing all the student work, how might you plan a small mini-unit to dig into the misconceptions and help students explore area and perimeter to develop a deeper level of understanding? Which misconceptions should be dealt with explicitly? Which misconceptions should be dealt with through investigation and direct experience? 74

30 Student 1 75

31 Student 2 Student 3 76

32 Student 4 Student 5 77

33 Student 6 Student 7 78

34 Student 8 Can you figure out what the student is calculating in part 1. In 2b the student gives the perimeter for the correct rectangle, but no dimensions. What do you think the student understands? 79

35 Student 9 Student 10 80

36 Student 11 Student 12 81

37 Ideas for Action Research 2 Problem of the Month One interesting task to help students stretch their thinking about 3-dimensional shapes is the problem of the month: Surrounded and Covered, from the Noyce Website: Ask students to work individually or in teams to solve the problem. Have them make posters of calculations they made, their conclusions, and graphics or visuals to support their thinking. The poster might also include other ideas they want to explore or conjectures they haven t had time to test. The purpose is to give them some complex mathematical thinking, that requires persistence, willingness to make mistakes, edit and revise, and is worth understanding the thinking of others. Now give students a chance to investigate part D of this task. 82

38 Reflecting on the Results for Sixth Grade as a Whole: Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? If you were to describe one or two big ides to take away and use for planning for next year, what would they be? What are some of the qualities that you saw in good work or strategies used by good students that you would like to help other students develop? Five areas that stand out for the Collaborative as a whole are: 1. Making comparisons Students have difficulty understanding what makes a good comparison mathematically. In Snail s Pace students often made a logical argument or gave a rate for the fastest snail, ignoring the speeds of the other snails. Students might have made a logical comparison between the two top speeds without converting them into equivalent units like inches per hour or inches per 20 minutes. Students did not understand that a good comparison would have used a similar unit for all the snails. In Area and Perimeter students did not think about finding the area or perimeter of the new rectangles so that both could be compared. Many just stated the prompt, such as same area with larger perimeter, with no quantification. Others just gave dimensions with no discussion or back up for how that completed the argument. 2. Understanding academic language of logic and probability Students did not understand the language or logic of a negative statement, such as not black. While this type of statement is not often used in everyday language, it is a typical form of statement in mathematics. 3. Making sense of all the relevant features of a pattern Students had trouble locating all the important features in A Number Pattern. Students might have thought of multiples, if they only looked at row five. Some students didn t notice that every row started and ended with a one. Some students didn t notice that the middle number(s) were the largest. Some students knew to add the above numbers to get the next row, 83

39 but didn t spend enough time noticing where the sum went in the row below. Some students were looking for the simplest possible patterns, like a sequence of even and odd numbers. Students need to develop the habit of mind to test their conjectures against all available evidence. 4. Seeing relationships between different representations of numbers Students did not know what to do with decimals that did not come out evenly at the hundredths place. They didn t understand that rounding gives an approximate answer but not the same or equal value as the original. Students did not know how to locate fractions on the number line. Students were often ordering fractions by the size of the denominator: 3/4, 2/5, then 3/8. 5. Making and testing conjectures Students had difficulty testing conjectures about area and perimeter. Students often neglected some of the constraints. Students didn t calculate the areas and perimeters to compare back to the original rectangles and justify that the conjecture had been fully met. Students didn t test different size rectangles before trying to write a generalization. 84

40 Examining the Ramp: Looking at Responses of the Early 4 s (31-33) The ramps for the sixth grade test: Snail Pace Part 4 Making a comparison using equal rates o Fixing the distance to compare times o Fixing the time to compare distance o Interpreting calculations to find the fastest rate Black and White - Part 2b Understanding a negative probability Number Pattern - Part 3b and 4 explaining patterns in words o Recognizing a doubling pattern o Looking at a pattern growing by increasing consecutive numbers Percent Cards Part D Understanding equivalence o Division with decimals beyond hundredths o Understanding that rounding gives an approximate value not an equivalent value Part 2- Locating fractions on a number line o Seeing relationships between previous work and the values to be placed on the number line o Understanding relative size of fractions Area and Perimeter Part 2 Keeping area the same and changing the perimeter o Conducting an investigation o Quantifying values of perimeter and area to verify conjecture With a group of colleagues look at student work around points. Use the papers provided or pick some from your own students. How are students performing on the ramp? What things impressed you about their performance? What are skills or ideas they still need to work on? Are students relying on previous arithmetic skills rather than moving up to more grade level appropriate strategies? What was missing that you would hope to see from students working at this level? When you read their words, do you have a sense of them understanding ideas personally or does it sound more like they are parroting things they ve heard the teacher say? How do you help students at this level step up their performance or see a standard to aim for in explaining their thinking? Are our expectations high enough for these students? For each response, can you think of some way that it could be improved? How do we provide models to help these students see how their work can be improved or what they are striving for? Do you think errors were caused by lack of exposure to ideas or misconceptions? Grade

41 Student 1 What would a student need to fix or correct their errors? What is missing to make it a top-notch response? What concerns you about their work? What strategies did you see that might be useful to show to the whole class? Grade

42 Student 1, part 2 Student 2 Grade

43 Student 2, part 2 Grade

44 Student 3 Grade

45 Student 3, part 2 Student 4 Grade

46 Student 4, part 2 Grade

47 Student 4, part 3 Student 5 Grade

48 Student 5, part 2 Grade

49 Student 5, part 3 Grade

Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 inches. 24 inches

Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 inches. 24 inches Winter Hat This problem gives you the chance to: calculate the dimensions of material needed for a hat use circle, circumference and area, trapezoid and rectangle Marie has a winter hat made from a circle,

More information

Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 inches

Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 inches Winter Hat This problem gives you the chance to: calculate the dimensions of material needed for a hat use circle, circumference and area, trapezoid and rectangle Marie has a winter hat made from a circle,

More information

Parallelogram. This problem gives you the chance to: use measurement to find the area and perimeter of shapes

Parallelogram. This problem gives you the chance to: use measurement to find the area and perimeter of shapes Parallelogram This problem gives you the chance to: use measurement to find the area and perimeter of shapes 1. This parallelogram is drawn accurately. Make any measurements you need, in centimeters, and

More information

G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

More information

High School Functions Building Functions Build a function that models a relationship between two quantities.

High School Functions Building Functions Build a function that models a relationship between two quantities. Performance Assessment Task Coffee Grade 10 This task challenges a student to represent a context by constructing two equations from a table. A student must be able to solve two equations with two unknowns

More information

Performance Assessment Task Parking Cars Grade 3. Common Core State Standards Math - Content Standards

Performance Assessment Task Parking Cars Grade 3. Common Core State Standards Math - Content Standards Performance Assessment Task Parking Cars Grade 3 This task challenges a student to use their understanding of scale to read and interpret data in a bar graph. A student must be able to use knowledge of

More information

Performance Assessment Task Cindy s Cats Grade 5. Common Core State Standards Math - Content Standards

Performance Assessment Task Cindy s Cats Grade 5. Common Core State Standards Math - Content Standards Performance Assessment Task Cindy s Cats Grade 5 This task challenges a student to use knowledge of fractions to solve one- and multi-step problems with fractions. A student must show understanding of

More information

1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I?

1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I? Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.

More information

Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table. Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

More information

Performance Assessment Task Leapfrog Fractions Grade 4 task aligns in part to CCSSM grade 3. Common Core State Standards Math Content Standards

Performance Assessment Task Leapfrog Fractions Grade 4 task aligns in part to CCSSM grade 3. Common Core State Standards Math Content Standards Performance Assessment Task Leapfrog Fractions Grade 4 task aligns in part to CCSSM grade 3 This task challenges a student to use their knowledge and understanding of ways of representing numbers and fractions

More information

Performance Assessment Task Which Shape? Grade 3. Common Core State Standards Math - Content Standards

Performance Assessment Task Which Shape? Grade 3. Common Core State Standards Math - Content Standards Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to

More information

Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations.

Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations. Performance Assessment Task Squares and Circles Grade 8 The task challenges a student to demonstrate understanding of the concepts of linear equations. A student must understand relations and functions,

More information

Performance Assessment Task Picking Fractions Grade 4. Common Core State Standards Math - Content Standards

Performance Assessment Task Picking Fractions Grade 4. Common Core State Standards Math - Content Standards Performance Assessment Task Picking Fractions Grade 4 The task challenges a student to demonstrate understanding of the concept of equivalent fractions. A student must understand how the number and size

More information

Students have to guess whether the next card will have a higher or a lower number than the one just turned.

Students have to guess whether the next card will have a higher or a lower number than the one just turned. Card Game This problem gives you the chance to: figure out and explain probabilities Mrs Jakeman is teaching her class about probability. She has ten cards, numbered 1 to 10. She mixes them up and stands

More information

High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable. Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations

More information

Total Student Count: 3170. Grade 8 2005 pg. 2

Total Student Count: 3170. Grade 8 2005 pg. 2 Grade 8 2005 pg. 1 Total Student Count: 3170 Grade 8 2005 pg. 2 8 th grade Task 1 Pen Pal Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Convert cake baking temperatures

More information

Geometry Solve real life and mathematical problems involving angle measure, area, surface area and volume.

Geometry Solve real life and mathematical problems involving angle measure, area, surface area and volume. Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find

More information

N Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to

More information

Shape Hunting This problem gives you the chance to: identify and describe solid shapes

Shape Hunting This problem gives you the chance to: identify and describe solid shapes Shape Hunting This problem gives you the chance to: identify and describe solid shapes Detective Sherlock Shapehunter tracks down solid shapes using clues provided by eyewitnesses. Here are some eyewitness

More information

How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old.

How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old. How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem Will is w years old. Ben is 3 years older. 1. Write an expression, in terms of

More information

E B F C 17 A D A D 16 B A G C 11

E B F C 17 A D A D 16 B A G C 11 Mystery Letters This problem gives you the chance to: form and solve equations A A A A 8 E B F C 17 A D A D 16 B A G C 11 9 11 14 18 In this table, each letter of the alphabet represents a different number.

More information

Cindy s Cats. Cindy has 3 cats: Sammy, Tommy and Suzi.

Cindy s Cats. Cindy has 3 cats: Sammy, Tommy and Suzi. Cindy s Cats This problem gives you the chance to: solve fraction problems in a practical context Cindy has 3 cats: Sammy, Tommy and Suzi. 1. Cindy feeds them on Cat Crunchies. Each day Sammy eats 1 2

More information

Granny s Balloon Trip

Granny s Balloon Trip Granny s Balloon Trip This problem gives you the chance to: represent data using tables and graphs On her eightieth birthday, Sarah s granny went for a trip in a hot air balloon. This table shows the schedule

More information

Number Factors. Number Factors Number of factors 1 1 1 16 1, 2, 4, 8, 16 5 2 1, 2 2 17 1, 17 2 3 1, 3 2 18 1, 2, 3, 6, 9, 18 6 4 1, 2, 4 3 19 1, 19 2

Number Factors. Number Factors Number of factors 1 1 1 16 1, 2, 4, 8, 16 5 2 1, 2 2 17 1, 17 2 3 1, 3 2 18 1, 2, 3, 6, 9, 18 6 4 1, 2, 4 3 19 1, 19 2 Factors This problem gives you the chance to: work with factors of numbers up to 30 A factor of a number divides into the number exactly. This table shows all the factors of most of the numbers up to 30.

More information

Performance Assessment Task Peanuts and Ducks Grade 2. Common Core State Standards Math - Content Standards

Performance Assessment Task Peanuts and Ducks Grade 2. Common Core State Standards Math - Content Standards Performance Assessment Task Peanuts and Ducks Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be fluent with addition

More information

A REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Performance Assessment Task Magic Squares Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures with algebraic

More information

8 th Grade Task 2 Rugs

8 th Grade Task 2 Rugs 8 th Grade Task 2 Rugs Student Task Core Idea 4 Geometry and Measurement Find perimeters of shapes. Use Pythagorean theorem to find side lengths. Apply appropriate techniques, tools and formulas to determine

More information

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Contents by Grade Level: Overview of Exam Grade Level Results Cut Score and Grade History

More information

Grade 5 Math Content 1

Grade 5 Math Content 1 Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

More information

Performance Assessment Task Bikes and Trikes Grade 4. Common Core State Standards Math - Content Standards

Performance Assessment Task Bikes and Trikes Grade 4. Common Core State Standards Math - Content Standards Performance Assessment Task Bikes and Trikes Grade 4 The task challenges a student to demonstrate understanding of concepts involved in multiplication. A student must make sense of equal sized groups of

More information

This problem gives you the chance to: calculate and interpret mean, medium and mode in a given table of realistic data

This problem gives you the chance to: calculate and interpret mean, medium and mode in a given table of realistic data Suzi s Company This problem gives you the chance to: calculate and interpret mean, medium and mode in a given table of realistic data Suzi is the chief executive of a small company, TechScale, which makes

More information

High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

High School Algebra Reasoning with Equations and Inequalities Solve systems of equations. Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student

More information

Operations and Algebraic Thinking Represent and solve problems involving multiplication and division.

Operations and Algebraic Thinking Represent and solve problems involving multiplication and division. Performance Assessment Task The Answer is 36 Grade 3 The task challenges a student to use knowledge of operations and their inverses to complete number sentences that equal a given quantity. A student

More information

Common Core State Standards. Standards for Mathematical Practices Progression through Grade Levels

Common Core State Standards. Standards for Mathematical Practices Progression through Grade Levels Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for

More information

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 % Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

More information

Grade Level Year Total Points Core Points % At Standard %

Grade Level Year Total Points Core Points % At Standard % Performance Assessment Task Marble Game task aligns in part to CCSSM HS Statistics & Probability Task Description The task challenges a student to demonstrate an understanding of theoretical and empirical

More information

Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

More information

Grade Level Year Total Points Core Points % At Standard %

Grade Level Year Total Points Core Points % At Standard % Performance Assessment Task Vincent s Graphs This task challenges a student to use understanding of functions to interpret and draw graphs. A student must be able to analyze a graph and understand the

More information

PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis* Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

More information

Finding Triangle Vertices

Finding Triangle Vertices About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

More information

Statistics and Probability Investigate patterns of association in bivariate data.

Statistics and Probability Investigate patterns of association in bivariate data. Performance Assessment Task Scatter Diagram Grade 9 task aligns in part to CCSSM grade 8 This task challenges a student to select and use appropriate statistical methods to analyze data. A student must

More information

Each plank of wood measures 1 inch by 9 inches by 48 inches. Each brick measures 3 inches by 4.5 inches by 9 inches.

Each plank of wood measures 1 inch by 9 inches by 48 inches. Each brick measures 3 inches by 4.5 inches by 9 inches. Shelves This problem gives you the chance to: solve problems in a spatial context identify and distinguish the four point graphs related to this situation Pete is making a bookcase for his books and other

More information

Balanced Assessment Test Algebra 2008

Balanced Assessment Test Algebra 2008 Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful

More information

Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. MP.

Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. MP. Performance Assessment Task Incredible Equations Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be able to understand

More information

Performance Assessment Task Fair Game? Grade 7. Common Core State Standards Math - Content Standards

Performance Assessment Task Fair Game? Grade 7. Common Core State Standards Math - Content Standards Performance Assessment Task Fair Game? Grade 7 This task challenges a student to use understanding of probabilities to represent the sample space for simple and compound events. A student must use information

More information

Mathematics Success Grade 8

Mathematics Success Grade 8 T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,

More information

Problem of the Month. Squirreling it Away

Problem of the Month. Squirreling it Away The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

More information

Second Grade Mars 2009 Overview of Exam. Task Descriptions. Algebra, Patterns, and

Second Grade Mars 2009 Overview of Exam. Task Descriptions. Algebra, Patterns, and Second Grade Mars 2009 Overview of Exam Task Descriptions Core Idea Task Algebra, Patterns, and Same Number, Same Shape Functions The task asks students to think about variables and unknowns in number

More information

Content standard focus: 5.MD Understand concepts of volume and relate volume to multiplication and addition.

Content standard focus: 5.MD Understand concepts of volume and relate volume to multiplication and addition. MP3, Grade 5 Task: Filling Boxes Practice standard focus: Mathematically proficient students at the elementary grades construct mathematical arguments that is, explain the reasoning underlying a strategy,

More information

Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation

Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units,

More information

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

More information

Mathematics Navigator. Misconceptions and Errors

Mathematics Navigator. Misconceptions and Errors Mathematics Navigator Misconceptions and Errors Introduction In this Guide Misconceptions and errors are addressed as follows: Place Value... 1 Addition and Subtraction... 4 Multiplication and Division...

More information

Performance Assessment Task Gym Grade 6. Common Core State Standards Math - Content Standards

Performance Assessment Task Gym Grade 6. Common Core State Standards Math - Content Standards Performance Assessment Task Gym Grade 6 This task challenges a student to use rules to calculate and compare the costs of memberships. Students must be able to work with the idea of break-even point to

More information

Area, Perimeter, Surface Area and Change Overview

Area, Perimeter, Surface Area and Change Overview Area, Perimeter, Surface Area and Change Overview Enduring Understanding: (5)ME01: Demonstrate understanding of the concept of area (5)ME02: Demonstrate understanding of the differences between length

More information

MARS Tasks Grade 6. NP=Number Properties NO=Number Operations PFA=Patterns Functions Algebra GM=Geometry & Measurement DA=Data Analysis

MARS Tasks Grade 6. NP=Number Properties NO=Number Operations PFA=Patterns Functions Algebra GM=Geometry & Measurement DA=Data Analysis MARS Tasks Grade 6 Page Name of MARS Task Year Math Strand Notes * Baseball Players 00 PS Mean, median, range in context * Gym 00 PFA, NO Analyze gym costs to solve problems * Square Elk 00 GM Find area,

More information

Grade 6 FCAT 2.0 Mathematics Sample Answers

Grade 6 FCAT 2.0 Mathematics Sample Answers Grade FCAT. Mathematics Sample Answers This booklet contains the answers to the FCAT. Mathematics sample questions, as well as explanations for the answers. It also gives the Next Generation Sunshine State

More information

Bedford Public Schools

Bedford Public Schools Bedford Public Schools Grade 4 Math The fourth grade curriculum builds on and extends the concepts of number and operations, measurement, data and geometry begun in earlier grades. In the area of number

More information

Grade 6 FCAT 2.0 Mathematics Sample Answers

Grade 6 FCAT 2.0 Mathematics Sample Answers 0 Grade FCAT.0 Mathematics Sample Answers This booklet contains the answers to the FCAT.0 Mathematics sample questions, as well as explanations for the answers. It also gives the Next Generation Sunshine

More information

Annotated work sample portfolios are provided to support implementation of the Foundation Year 10 Australian Curriculum.

Annotated work sample portfolios are provided to support implementation of the Foundation Year 10 Australian Curriculum. Work sample portfolio summary WORK SAMPLE PORTFOLIO Annotated work sample portfolios are provided to support implementation of the Foundation Year 10 Australian Curriculum. Each portfolio is an example

More information

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;

More information

Division with Whole Numbers and Decimals

Division with Whole Numbers and Decimals Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers

More information

Planning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3

Planning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3 Mathematics Planning Guide Grade 6 Factors and Multiples Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_factorsmultiples/index.html

More information

Pythagorean Theorem. Overview. Grade 8 Mathematics, Quarter 3, Unit 3.1. Number of instructional days: 15 (1 day = minutes) Essential questions

Pythagorean Theorem. Overview. Grade 8 Mathematics, Quarter 3, Unit 3.1. Number of instructional days: 15 (1 day = minutes) Essential questions Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean Theorem Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Prove the Pythagorean Theorem. Given three side lengths,

More information

Ministry of Education. The Ontario Curriculum Exemplars Grade 6. Mathematics. Samples of Student Work: A Resource for Teachers

Ministry of Education. The Ontario Curriculum Exemplars Grade 6. Mathematics. Samples of Student Work: A Resource for Teachers Ministry of Education The Ontario Curriculum Exemplars Grade 6 Mathematics Samples of Student Work: A Resource for Teachers 2002 Contents Introduction......................................................

More information

Numbers and Operations in Base 10 and Numbers and Operations Fractions

Numbers and Operations in Base 10 and Numbers and Operations Fractions Numbers and Operations in Base 10 and Numbers As the chart below shows, the Numbers & Operations in Base 10 (NBT) domain of the Common Core State Standards for Mathematics (CCSSM) appears in every grade

More information

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

More information

Solving Equations with One Variable

Solving Equations with One Variable Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Solve linear equations in one variable

More information

Session 7 Fractions and Decimals

Session 7 Fractions and Decimals Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

More information

Performance Assessment Task Symmetrical Patterns Grade 4. Common Core State Standards Math - Content Standards

Performance Assessment Task Symmetrical Patterns Grade 4. Common Core State Standards Math - Content Standards Performance Assessment Task Symmetrical Patterns Grade 4 The task challenges a student to demonstrate understanding of the concept of symmetry. A student must be able to name a variety of two-dimensional

More information

Number Talks. 1. Write an expression horizontally on the board (e.g., 16 x 25).

Number Talks. 1. Write an expression horizontally on the board (e.g., 16 x 25). Number Talks Purposes: To develop computational fluency (accuracy, efficiency, flexibility) in order to focus students attention so they will move from: figuring out the answers any way they can to...

More information

Mathematics. What to expect Resources Study Strategies Helpful Preparation Tips Problem Solving Strategies and Hints Test taking strategies

Mathematics. What to expect Resources Study Strategies Helpful Preparation Tips Problem Solving Strategies and Hints Test taking strategies Mathematics Before reading this section, make sure you have read the appropriate description of the mathematics section test (computerized or paper) to understand what is expected of you in the mathematics

More information

Algorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure

Algorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm

More information

Multiplying Fractions by a Whole Number

Multiplying Fractions by a Whole Number Grade 4 Mathematics, Quarter 3, Unit 3.1 Multiplying Fractions by a Whole Number Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Apply understanding of operations

More information

Problem of the Month: Perfect Pair

Problem of the Month: Perfect Pair Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

More information

Understanding Place Value of Whole Numbers and Decimals Including Rounding

Understanding Place Value of Whole Numbers and Decimals Including Rounding Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned

More information

High School Functions Interpreting Functions Understand the concept of a function and use function notation.

High School Functions Interpreting Functions Understand the concept of a function and use function notation. Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.

More information

LESSON 10 GEOMETRY I: PERIMETER & AREA

LESSON 10 GEOMETRY I: PERIMETER & AREA LESSON 10 GEOMETRY I: PERIMETER & AREA INTRODUCTION Geometry is the study of shapes and space. In this lesson, we will focus on shapes and measures of one-dimension and two-dimensions. In the next lesson,

More information

Supporting your child with maths

Supporting your child with maths Granby Primary School Year 5 & 6 Supporting your child with maths A handbook for year 5 & 6 parents H M Hopps 2016 G r a n b y P r i m a r y S c h o o l 1 P a g e Many parents want to help their children

More information

CARMEL CLAY SCHOOLS MATHEMATICS CURRICULUM

CARMEL CLAY SCHOOLS MATHEMATICS CURRICULUM GRADE 4 Standard 1 Number Sense Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers 1 and decimals relate to simple fractions. 4.1.1 Read and write

More information

Grade 5 Measurement: Additional Sample 1 Planning a Backyard

Grade 5 Measurement: Additional Sample 1 Planning a Backyard Grade 5 Measurement: Additional Sample 1 Planning a Backyard Context This class has had previous practice estimating and measuring the area and perimeter of irregular shapes. Mathematical Concepts recognize

More information

Geometry: A Better Understanding of Area

Geometry: A Better Understanding of Area Geometry: A Better Understanding of Area 6G1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other

More information

Performance Based Learning and Assessment Task Confetti Task

Performance Based Learning and Assessment Task Confetti Task Performance Based Learning and Assessment Task Confetti Task I. ASSESSMENT TASK OVERVIEW & PURPOSE: In this task, Geometry students will investigate how surface area and volume is used to estimate the

More information

Measurement with Ratios

Measurement with Ratios Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

More information

4 th Grade. Math Common Core I Can Checklists

4 th Grade. Math Common Core I Can Checklists 4 th Grade Math Common Core I Can Checklists Math Common Core Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems. 1. I can interpret a multiplication equation

More information

22 Proof Proof questions. This chapter will show you how to:

22 Proof Proof questions. This chapter will show you how to: 22 Ch22 537-544.qxd 23/9/05 12:21 Page 537 4 4 3 9 22 537 2 2 a 2 b This chapter will show you how to: tell the difference between 'verify' and 'proof' prove results using simple, step-by-step chains of

More information

Progressing toward the standard

Progressing toward the standard Report Card Language: add, subtract, multiply, and/or divide to solve multi-step word problems. CCSS: 4.OA.3 Solve multistep work problems posed with whole numbers and having whole-number answers using

More information

Level Descriptors Maths Level 1-5

Level Descriptors Maths Level 1-5 Level Descriptors Maths Level 1-5 What is APP? Student Attainment Level Descriptors APP means Assessing Pupil Progress. What are the APP sheets? We assess the children in Reading, Writing, Speaking & Listening,

More information

Pupils analyse a numerical puzzle, solve some examples and then deduce that a further example is impossible.

Pupils analyse a numerical puzzle, solve some examples and then deduce that a further example is impossible. Task description Pupils analyse a numerical puzzle, solve some examples and then deduce that a further example is impossible. Suitability National Curriculum levels 5 to 7 Time Resources 30 minutes to

More information

Archdiocese of Washington Catholic Schools Academic Standards Mathematics

Archdiocese of Washington Catholic Schools Academic Standards Mathematics 5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,

More information

MATH Fundamental Mathematics II.

MATH Fundamental Mathematics II. MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

More information

5 th Grade Mathematics

5 th Grade Mathematics 5 th Grade Mathematics Instructional Week 20 Rectilinear area with fractional side lengths and real-world problems involving area and perimeter of 2-dimensional shapes Paced Standards: 5.M.2: Find the

More information

Mathematics Content Strands

Mathematics Content Strands Grade 6 Mathematics: Pre-Kindergarten Through Grade 8 Mathematics Content Strands M1 Numbers and Operations Number pervades all areas of mathematics. The other four Content Standards as well as all five

More information

Planning Guide. Number Specific Outcomes 8, 9, 10 and 11

Planning Guide. Number Specific Outcomes 8, 9, 10 and 11 Mathematics Planning Guide Grade 5 Working with Decimal Numbers Number Specific Outcomes 8, 9, 10 and 11 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg5/html/pg5_workingwithdecimalnumbers/index.html

More information

Grade 4 Performance Task

Grade 4 Performance Task Planting Tulips 1. Classroom Activity 2. Student Task 3. Task Specifications 4. Scoring Rubric Classroom Activity Note: Since performance tasks span different parts of the assessment system (summative,

More information

Performance Assessment Task Baseball Players Grade 6. Common Core State Standards Math - Content Standards

Performance Assessment Task Baseball Players Grade 6. Common Core State Standards Math - Content Standards Performance Assessment Task Baseball Players Grade 6 The task challenges a student to demonstrate understanding of the measures of center the mean, median and range. A student must be able to use the measures

More information

#1 Make sense of problems and persevere in solving them.

#1 Make sense of problems and persevere in solving them. #1 Make sense of problems and persevere in solving them. 1. Make sense of problems and persevere in solving them. Interpret and make meaning of the problem looking for starting points. Analyze what is

More information

RIT scores between 191 and 200

RIT scores between 191 and 200 Measures of Academic Progress for Mathematics RIT scores between 191 and 200 Number Sense and Operations Whole Numbers Solve simple addition word problems Find and extend patterns Demonstrate the associative,

More information

Robyn Seifert Decker

Robyn Seifert Decker Robyn Seifert Decker UltraMathPD@gmail.com Place Value Addition Subtraction Problem Solving Fractions If time allows: Multiplication and Division Spiral of change From Prochaska, DiClemente & Norcross,

More information

Number & Place Value. Addition & Subtraction. Digit Value: determine the value of each digit. determine the value of each digit

Number & Place Value. Addition & Subtraction. Digit Value: determine the value of each digit. determine the value of each digit Number & Place Value Addition & Subtraction UKS2 The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value

More information