assessing student understanding - carnegie...

PARTICIPANT HANDOUTS

Assessing Student Understanding"Formative assessment" is defined in many different ways, but it primarily involves strategies to reveal information about student thinking that can be used to inform instruction. This session is interactive and collaborative. Participants will engage in activities created from Carnegie Learning text resources that to allow students to reveal their thinking.

Strategies to allow students to interact with the curriculum daily

Four Corners – Using a “ticket” in to select the corner with the matching choice.

My Favorite “No” – Collect student work and use mistakes to quickly clarify concepts.

Simulations Line Up – Students line up facing forward, person in back starts the response and passess forward to next person to enter the next response, continue unit it reaches the front. Person in front moves to the back.

Pass the Problem – In teams, students each writes a response on their own piece of paper. Then pass their papers clockwise so each teammate can add to the prior response.

Exit/Entrance Ticket - Response are given out the door or in the door with immediate feedback.

Group Work Placemat – Each student works in their designated area on the placemat. All four answers are added together and the sum is recorded in the middle. This makes it really easy for the teacher to see if the group is on track.

Write-Boards/Communicator Boards – Put a piece of colored paper into a sheet protector and you’ve got a very cheap white-board! Add a marker and students can record their work and hold up their answers!!

Coach/Player – Pair up and assign roles (one student is the coach and one is the player). Round 1 the coach puts his/her pencil down and “coaches” (or tells) the player how to solve a problem…..the player records the work. Round 2 roles are reversed and the process is repeated. Continue until all problems are complete.

Think-Pair-Share – When you want participants to talk about something, give them a set time to think individually, then pair with another participant and share their thoughts. You can repeat having them find different partners as well.

Mix –Pair – Share – The class “mixes” until the teacher calls “pair”. Students find a new partner to discuss an answer or teacher question.

Stand-Up-Hand-Up-Pair-Up - students will pair up by quickly finding a partner with their hand raised.

Fan and Pick – Problem cards or check for understanding questions are given to a group to pick and answer.

Vocabulary Knowledge Rating (VKR) - One indication that students have a deep understanding of a mathematical topic is their ability to use essential vocabulary when they speak and write about that topic. The Vocabulary Knowledge Rating uses a double-assessment process to help students build their mastery of critical terms and concepts.

Knock Out – Team game to promote math fluency and accuracy. Teams/pairs compete to answer math questions in a specified amount of time. Correct answers will have the chance to “knock out” other teams.

Quiz Quiz Trade - Quiz a partner, get quizzed by a partner, and then trade cards to repeat the process with a new partner.

Inside-Outside Circle – Students rotate in concentric circles to face new partners for sharing, quizzing, or problem solving.

Frayer Model – Students are asked to provide a Definition of the word, Facts or Characteristics of the word, Examples, and Nonexamples/Counterexamples.

NOTES:

Student Work Task

Making PunchSCENARIO – Each year, your class presents its mathematics portfolio to parents and community members. This year, your homeroom is in charge of the refreshments for the reception that follows the presentations.

Four students give their recipes for the lemon-lime punch. The class decides to analyze the recipes to determine which one will make the fruitiest-tasking punch. The recipes are shown below.

Adam’s Recipe Bobbi’s Recipe4 cups lemon-lime concentrate 3 cups lemon-lime concentrate8 cups club soda 5 cups club soda

Carlos’ Recipe Zeb’s Recipe2 cups lemon-lime concentrate 1 cup lemon-lime concentrate3 cups club soda 4 cups club soda

Determine a solution on your own first. Share your methods with your table group.

Reflection Planning Questions:

1. What are all the MATH SKILLS needed for a student to solve this problem? Are there different levels of understanding? (List the explicit, implicit, and prerequisite skills)

What does the Standard(s) look like for the different levels of understanding?

2. Provide a definition for each “assessment rating” below before looking at student examples. What would you expect to see/hear from a student in the classroom based on the RP standards you choose?

1 – Limited

2 – Proficient

3 – Advanced

1

CCSS: Ratios and Proportional Relationships 6.RPUnderstand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

c. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing

CCSS: Ratios and Proportional Relationships 7.RPAnalyze proportional relationships and use them to solve real-world and mathematical problems.

1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

2. Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for

equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent

Review the four Student Work Samples that follow and discuss the questions below with your table mates.

TEI Student Work Group Discussion

1. What methods did each group use to determine which punch was the fruitiest?

2. How would you rate each student group’s understanding of the standard? (use your definition for limited, proficient, advanced)

3. How would you facilitate the discussion from the gallery walk of student work?

4. How would you facilitate any misconceptions?

Student Group 1:

Student Group 2:

Student Group 3:

Student Group 4

:

Folded Frayer Model Directions

Hold a sheet of 8 ½ x 11 inch paper like a portrait. Then fold the sheet in half horizontally.

Now fold the paper in half vertically to create 4 sections if you open it up.

On the corner where the folds meet, fold a right triangle with the bottom edge of the triangle parallel to the bottom edge of the paper.

Now open the paper flat, put the word in the center diamond, and label the four sections like they are on a regular Frayer Model.

Definition Facts/Char-acteristics

Examples Nonexamples

cot A =

cot F =

cot A =

cot A =

Find x

Find x

A boat travels in the following path. How far north did it travel?

During a group hike, a park ranger makes the following path. How far west did they travel?

A surveyor makes the following diagram of a hill. What is the height of the hill?

To find the height of a tree, a botanist makes the following diagram. What is the height of the tree?

Calculate the measure of angle X

Calculate the measure of angle X

Multiple Representation Organizer

Verbal TableSummary of scenario:

Define the variables:

Independent:

Dependent:

Constants:

Variable Quantities

Units

Graph Equation/AnalysisCreate a graph that represents the information in the table.

Write an Equation(s):

1. What does the slope mean in context to the problem scenario?

2. What does the y-intercept represent in context to the problem scenario?

Names: _________________________

Verbal Table

UnitedComm charges $0.25 per minute to call anywhere in the world with a monthly fee of $15.

Define the variables:Independent:

Dependent:

Constants:

Number of Minutes

Cost of Call

0

10

20

30

40

50

60


1. Write an Equation:

2. How much would it cost to talk for 35 minutes?

3. How much would it cost to talk for n minutes?

4. How many minutes would you have talked for if the call cost $70.00

Name: ______________________________

Verbal TableGlobalComm charges $0.35 per minute to call anywhere in the world.

0


Dependent:

Constants:

10

20

30

40

50

60

Graph AnalysisCreate a graph that represents the information in the table.





Name_____________________________________

Verbal Table

UpperLimit charges $0.15 per minute to call anywhere in the world with a monthly fee of $35.


Dependent:

Constants:

0

10

20

30

40

50

60






Name_______________________________________

Verbal TableFlatWorld charges $0.20 per minute to call anywhere in the world with a monthly fee of $30.


Dependent:

Constants:

0

10

20

30

40

50

60






Name__________________________________________

Verbal TableTeleLink charges $0.10 per minute to call anywhere in the world with a monthly fee of $40.


Dependent:

Constants:

0

10

20

30

40

50

60






Vocabulary Knowledge Rating (VKR)

One indication that students have a deep understanding of a mathematical topic is their ability to use essential vocabulary when they speak and write about that topic. The Vocabulary Knowledge Rating uses a double-assessment process to help students build their mastery of critical terms and concepts.

Process Steps:

1. Introduce students to critical vocabulary they will encounter in the lesson or unit by reading the words out loud.

2. Have the students rate their knowledge of each word by selecting the appropriate number from the four point scale. When finished, have the students add up all the numbers and compute their knowledge rating.

3. Over the course of the lesson or unit, have the students reflect on their initial ratings and how their understanding of the vocabulary has changed.

4. After completing the lesson or unit, have the students again rate their knowledge of the vocabulary and compare this new rating to their initial rating.

You can create an electronic form as well.o Google Forms

9.2 Angles and More Angles

TERMS I know and can explain the term

I think I knowthe term

I have seen or heard of the term

Never heard of the term

ray 4 3 2 1

angle 4 3 2 1

sides of an angle 4 3 2 1

vertex 4 3 2 1

degrees 4 3 2 1

acute angle 4 3 2 1

right angle 4 3 2 1

obtuse angle 4 3 2 1

straight angle 4 3 2 1

congruent angles 4 3 2 1

bisect 4 3 2 1

angle bisector 4 3 2 1

PRE-Knowledge Rating Date

9.2 Angles and More Angles





ray 4 3 2 1

angle 4 3 2 1

sides of an angle 4 3 2 1

vertex 4 3 2 1

degrees 4 3 2 1

acute angle 4 3 2 1

right angle 4 3 2 1

obtuse angle 4 3 2 1

straight angle 4 3 2 1

congruent angles 4 3 2 1

bisect 4 3 2 1

angle bisector 4 3 2 1

POST-Knowledge Rating Date

Lesson / Chapter / Unit Title





4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

PRE-Knowledge Rating Date

Lesson / Chapter / Unit TITLE





4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

4 3 2 1

POST-Knowledge Rating Date

What is the definition of Scaling Down?

When you divide the numerator and the denominator by the same factor.

What is the difference between a rate and a ratio?

A rate compares two quantities that are different. A ratio compares to quantities that are the same.

What method would you use to solve the proportion 14 trees

3 rows= ? tress

9 rows

Scale up by 3, because the number of rows has increase by 3 times the given ratio, so there will be three times more trees in 9 rows.

What is the definition of Scaling Up?

When you multiple the numerator and the denominator by the same factor.

How do you know a number is divisible by 3?

If the sum of the digits is divisible by 3, the original number is divisible by 3.Ex. 245 2+4+5=11, NOEx. 324 3+2+4=9, YES


What rule can you use?

Check the last 2 digits of the number, if they are divisible by 4, then the number is divisible by 4.Ex. 2052, divide 52 by 4, it equals 13, so 2052 is divisible by 4.

How do you know if a number is divisible by 5?

What rule can you use?

If a number ends in zero or five, then the number is divisible by 5.


What rule or rules do I

If a number is divisible by 2 and 3, then it is also divisible by 6.Ex. 54 2?:even yes, 3?:5+4=9, yes YES 54 is by 6

need to know? Ex. 86 2? YES 3?:8+6=14 NO

How do you know a number is divisible by

8?

Check the last three digits of the number, if they are divisible by 8, then the original number is divisible by 8.


9?

Add all the digits together, if the sum is divisible by 9 then the original number is divisible by 9.Ex. 9,787 9+7+8+7=31 NOEx. 261 2+6+1=9 YES


10?

Any number that ends in a zero is divisible by 10.


2?

Any even number is divisible by 2.

(the number ends in 0, 2, 4, 6, 8)

Find the least common multiple (LCM) and greatest common factor (GCF) of 45 and 9.

LCM: 45GCF: 9

justifyFind the least common multiple (LCM) and greatest common factor (GCF) of 18 and 4.

LCM: 36GCF: 2

justify

Find the least common multiple (LCM) and greatest common factor (GCF) of 20 and 15.

LCM: 60GCF: 5

justify

How many minutes are There are 60 minutes in

in one hour?

How many minutes are in three hours?Explain your answer.

1 hour.

There are 180 minutes in 3 hours. I scaled up by a factor of 3. (60x3=180)

What is the difference between a unit rate and a rate?

A unit rate has a denominator of 1 unit.EX. 3 steps/1 min.A rate has a denominator other than one. EX. 36 steps/9 min.

What do you need to multiply 8 by to get to

24?

3, three

8x3 = 2424÷8 = 3

What do you need to multiply 6 by to get to

42?

7, seven

6x7 = 4242÷6 = 7

Becky can swim 4 laps in her pool in 10 minutes. How many laps will she swim in one hour if she stays at the same rate?EXPLAIN your method.

4 laps10min.

= ? laps60min

Scale up by 64x6 = 24 laps

Becky will swim 24 laps in one hour.

Determine the unit rate.

$0.40 $1.604 pens 16 pens

Each pen is $0.10or

$0.10 per pen

When you write a proportion, how do you know what quantities to write in the numerators and denominators in a

ratio?

The quantities in the numerator and

denominator of the proportions have to be the same in each of the

ratios.

There are 18 people sitting at 6 tables, how many are

each table?186

=1?

What is wrong with the proportion?

The ratios are not set up with the same quantities

on the numerator and the denominator.

18 people6 tables

=? people1 table

How can you determine if two ratios are proportional?

If two ratios are proportional then I can scale up or scale down by the same factor. I can also use the means and extremes method or cross

products are equal.

A pack of 40 AAA batteries costs $25.95. Explain how

to find the cost for one AAA battery.

Set up a proportion with two equivalent ratios. Solve to find the cost for one AAA

battery.

Ask for an example and verify the example

A package of 24 rolls of toilet paper costs $16.25.

Explain how to find the cost of one roll of toilet

Set up a proportion with two equivalent ratios. Solve to find the cost for one roll of

toilet paper.

paper. Ask for an example and verify the example

What is the inverse operation for division?

What is the definition of inverse operation?

Multiplication and division are inverse operations – multiplication “undoes” divisionAddition and subtraction are inverse operations – they “undo” each other

What is the definition of variable?

Why do we use variables?

A variable is a letter or symbol used to

represent a number. We often use variables to

represent a number we do not know the value for.

Most of these questions/answers are from Course 2 Chapter 1 questions in the TIG margin and problems worked on in the chapter and review skills to practice.

QUESTION on right Column – RESPONSE on Left Column

assessing student understanding - carnegie...

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