1. Supporting Rigorous Mathematics Teaching and Learning
Using Assessing and Advancing
Questions to Target Essential Understandings
Overview of the Module:
Participants will consider a means of assessing and advancing student learning during instruction. We will engage in a lesson as adult learners. When we are engaging in the lesson, take note of the kinds of questions that I am asking you. We will not pretend we are students or think about how students will respond. Our goal is to deepen our understanding of the standards and to make sense of the use of models when working with the concept.
No Prior Knowledge Necessary.
Materials:
Slides with note pages
Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content).
Participant handouts
Chart paper
Markers
One-inch square tiles
Tennessee Department of Education
Elementary School Mathematics
Grade 3

2. Rationale
There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). By engaging in an analysis of a lesson planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.
Directions:
(SAY) Take a minute to read the rationale for the lesson. The quote comes from the Common Core State Standards.
As you can see from the rationale, basing instructional decisions on student thinking is critical. Teachers must listen to and attempt to make sense of what students are or are not understanding. We will engage in the lesson with the goal of deepening our understanding of the concept of fractions. As the teacher, I will attempt to understand your thinking.

3. Session Goals Participants will learn to set clear goals for a lesson;
learn to write essential understandings and consider the relationship to the CCSS; and
learn the importance of essential understandings (EUs) in writing focused advancing questions.
Directions:
Read the goals on the slide.

4. Overview of Activities
Participants will:
engage in a lesson and identify the mathematical goals of the lesson;
write essential understandings (EUs) to further articulate a standard;
analyze student work to determine where there is evidence of student understanding; and
write advancing questions to further student understanding of EUs.
Directions:
(SAY) We will engage in a task and then step out and reflect on our engagement in the task. We will consider how our learning was supported and which standards we had opportunities to think about and use when figuring out the solution path.

5. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as set up by the teachers
TASKS
as they appear in curricular/ instructional materials
TASKS
as implemented by students
Student Learning
Directions:
(Say) As you recall, the Mathematical Tasks Framework (TAG) shows the phases that tasks pass through. We are starting with a high-level task. As the task is enacted, 67% of the time tasks that are high-level decline or students do not have opportunities to engage in the high-level thinking that is possible. There are many reasons why this happens. Some include teacher wait time, using the task at the wrong time in student learning, and the teacher giving students steps for solving the problem. 
So if telling is not allowed, then we have to have something else to do as teachers in order to prevent us from telling. This is the use of assessing and advancing questions as students are exploring and making sense of the task. You may recall these from the summer, if you attended sessions. 
If not, do not worry because we are working on them again. This time, however, we getting a little more specific about the types of assessing and advancing questions that we should be asking students.
Stein, Smith, Henningsen, & Silver, 2000

6. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as set up by the teachers
TASKS
as they appear in curricular/ instructional materials
TASKS
as implemented by students
Student Learning
Directions:
(SAY) Asking assessing and advancing questions is only one of the instructional methods that we will be learning about in the two days that we are together. We will also learn about Accountable Talk moves and the movement between representations.
Stein, Smith, Henningsen, & Silver, 2000
Setting Goals
Selecting Tasks
Anticipating Student Responses
Orchestrating Productive Discussion
Monitoring students as they work, asking assessing and advancing questions
Selecting solution paths
Sequencing student responses
Connecting student responses via Accountable Talk® discussion
Accountable Talk ® is a registered trademark of the University of Pittsburgh

7. Solving and Discussing Solutions to the Half of a Whole Task
Directions:
(SAY) Let’s engage in the lesson.

8. The Structure and Routines of a Lesson
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small Group Problem Solving
Generate and Compare Solutions
Assess and advance Student Learning
Share Discuss and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
MONITOR: Teacher selects
examples for the Share, Discuss,
and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
SHARE: Students explain their
methods, repeat others’ ideas,
put ideas into their own words,
add on to ideas and ask
for clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
similarities and difference
between solution paths.
FOCUS: Discuss the meaning
of mathematical ideas in each
representation
REFLECT: Engage students
in a Quick Write or a discussion
of the process.
Set Up the Task
Set Up of the Task
Directions: (Optional)
(SAY) This graphic shows the phases that a high-level task goes through as it is enacted in a classroom. It assumes we have already selected a high-level task and are aware of how the task will help students work towards or use standards for mathematical practice as a means of understanding mathematical content.
You have seen this graphic many times. This lesson structure ensures that students have individual problem-solving time, small group problem-solving time, and whole group discussion time. All these opportunities are times when the teacher can assess student thinking. They are also times when students can find solution paths and practice strategies before they are shared with a larger group.
One of the most important phases of the lesson is the Explore Phase of the lesson. This is the time when students are making sense of the task and problem solving. What is the teacher doing during this phase of the lesson? Why do we say this is a time when the teacher can differentiate instruction for students?

9. Half of a Whole: Task Analysis
Solve the task. Write sentences to describe the mathematical relationships that you notice.
Anticipate possible student responses to the task.
Directions:
Read the directions.

10. Half of a Whole Task
Identify all of the figures that have one half shaded. Be prepared to show and explain how you know that one half of a figure is shaded. If a figure does not show one half shaded, explain why. Make math statements about what is true about a half of a whole.
Adapted from Watanabe, 1996
Directions:
Circulate asking assessing and advancing questions, as participants solve the task.
Assess: What made you call these figures showing halves?
Advance: How can this one be halves? It shows more than half shaded.
Assess: How do you know this one shows halves? [Diagonal figure (f)]
Advance: How can you prove that they are halves? You can fold, cut, or do what you need to do.
Assess: Look at figures (a) and (c). Which is more?
Advance: What can you do to prove that both of these show halves?
Assess: Does figure (g) show halves? How do you know?
Advance. What size piece is the shaded portion of the figure? What size piece is the unshaded portion?
Possible Solution Path and Probing Questions
How is figure (a) different from figure (e)?
How is figure (a) related to figure (c)? How is it different from figure (c)? Does it matter that the two pieces, the are in the middle of the figure? Why not?
Tell me about figure (g). Some people claim it shows halves and some people claim it does not show halves. Who can explain?
Tell me about (d). Does it show halves? How do you know? There are two ways of explaining that figure (d) shows halves. What are the two ways?
So we can talk about halves as equal amounts of area and we can talk about half of 6 as 3. Can someone else talk about two ways of proving that figure (a) shows halves?

11. Half of a Whole: Task Analysis
Study the Grade 3 CCSS for Mathematical Content within the Number and Operations—Fractions domain.
Which standards are students expected to demonstrate when solving the fraction task?
Identify the CCSS for Mathematical Practice required by the written task.
Directions:
Read the directions

12. The CCSS for Mathematical Content: Grade 3
Number and Operations—Fractions 3.NF
Develop understanding of fractions as numbers.
3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.A.2a
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.A.2b
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Facilitator Notes:
Which standards will students have opportunities to use when solving the Fraction of the Whole Task?
Possible Probing Questions and Possible Responses (italics):
Will students have opportunities to work on this standard 3.NF.A1? If so why?
Students will talk about as one part of the whole that is cut into two parts or 2 parts of the whole that is cut into four parts or 3 parts of the whole that is cut into 6 parts.
Will students have opportunities to work on this standard 3.NF.A2, A2a, A2b? Why or why not?
These standards involve work with a number line and this task does not require that students use a number line.
Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

13. The CCSS for Mathematical Content: Grade 3
Number and Operations—Fractions 3.NF
Develop understanding of fractions as numbers.
3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.A.3a
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.A.3b
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.A.3c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
3.NF.A.3d
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Possible Probing Questions and Possible Responses (italics):
When working on the Half of the Whole Task, students will have an opportunity to work on:
Developing an understanding of fractions as numbers.
Do students have opportunities to work on standard 3.NF.A3? If so, in what ways?
Students must explain how and show halves.
Do students have opportunities to work on standard 3.NF.A3a?
The teacher can press students to say how and show halves and the teacher can ask if the amounts are equivalent [equal] and why.
Do students have opportunities to work on standard 3.NF.A3b?
Students do not have to generate simple equivalent fractions, e.g., = Engagement in the task will result in a list of equivalent fractions. This standard calls for students to generate them numerically. The teacher can certainly extend the task and ask students for other shares of wholes that refer to halves, such as , ,
Do students have opportunities to work on standard 3.NF.A3c?
Students can discuss = 1 whole. One opportunity exists in this task for students to discuss standard 3.NF.A3c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = ; recognize that = 6; locate and 1 at the same point of a number line diagram.
Do students have opportunities to work on standard 3.NF.A3d?
Students can compare to and determine that the numerator is half of the denominator in both cases; therefore, they both refer to half of the figure. The teacher can ask students to use the symbols >, =, or <, to justify which is more, less, or equal to each other. The task itself does not require that students use this standard.
Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

14. The CCSS for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Let’s consider how the tasks relate to the Standards for Mathematical Practice:
Make sense of problems and persevere in solving them:
Students must make sense of which figures show halves. They must persevere because several figures must be considered.
Reason abstractly and quantitatively:
Students do not have to work on this standard.
Construct viable arguments and critique the reasoning of others:
Students must identify the figures that show halves and explain how they know they show halves.
Model with mathematics:
Students do not have to write equations. They are simply asked to identify the figures that show halves. They do not have to write other equivalent fractions. The teacher asks students who are ready for this level of work to write the fraction for the figures showing 𝟐 𝟒 and 𝟑 𝟔 .
Use appropriate tools strategically:
Students can be given the shapes and permitted to cut and fold the figures.
Attend to precision:
A precise explanation is expected.
Look for and make use of structure
Students must make sense of the meaning of a denominator and a numerator when solving this task. Students must realize that the halves shown on the square figures need to take up the same amount of space.
Look for and express regularity in repeated reasoning.
Students will have to use repeated reasoning when considering each of the figures in order to determine if halves are shown.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 14

15. The Common Core State Standards
Common Core State Standards for Mathematical Content and Mathematical Practice
Essential Understandings
Directions:
(SAY) This is a tool to organize the correlation between the CCSS (both content and practice) with the essential understandings that underlie the standards. We will take a look at a sample EU in a few moments.

16. Mathematical Essential Understanding Not All Halves Are Created Equal
3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
What is mathematically true about the figures shown above?
Probing Facilitator Questions and Possible Responses:
How are the standard, the objective, and the essential understanding different from each other?
The standard is the stated expectation. It is what the student is to understand, The standard is written using mathematical symbols.
The objective tells what the student will do and how the student will engage. Do we know what the student will understand?
The essential understanding is the mathematical truth that is the foundation of the standard and the objective. It does not state what a student must do. If this is not part of the teacher’s content knowledge, the depth of understanding will not be reached by the student.
What part of the EU statement is critical and why?
The reason why all of the pieces in the numerator get partitioned is the critical piece of this statement.
What is the added value of the essential understandings?
What is the value in knowing the essential understandings?
Objective
Students will discover, via the use of the fractional pieces, that not all halves are equal.
Essential Understanding
If the wholes differ, then a fractional piece from each of the wholes will not be equal because their initial whole was not the same (e.g., of a large pizza is not the same as of a small pizza).

17. Essential Understanding
Mathematical Essential Understanding A Whole Can Be Represented as a Fraction
3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
𝟒 𝟒
Objective
Students will recognize that fractions in the form 1 1 , 2 2 , 3 3 , 4 4 , etc., are fractional names for a whole, or 1.
Essential Understanding
Directions:
Group the participants in triads and assign page 9, 10, or 11 of the Participant Handout for each to develop.
What is the essential understanding related to this objective and standard?
If students know that 𝟒 𝟒 = 1, then what must they understand?
What might you hear third grade students say?
It is all of them because there are four pieces and all four are shaded.
What question might you ask students to prompt them to make a connection between the denominator and the numerator?

18. Essential Understanding
Mathematical Essential Understanding Half of the Denominator Is Half of the Whole
3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Objective
Students recognize that two equal parts of the whole represent half of the figure.
Essential Understanding
Directions:
Possible Probing Questions and Possible Responses:
What is the essential understanding related to this standard?
Why can you look at the denominator and immediately tell me what is half?
How do you know that 𝟑 𝟔 is half? What else would a student need to say in order for you to be convinced that he or she knows that splitting the denominator tells us half?
We would need to hear the student say = 6 or 6 divided by 2 is 3
What is the essential understanding if this is what a student must be able to say?
We can determine halves by looking at the denominator and determining if the numerator is half of the denominator. By splitting the denominator into two, we know it is half because two fair shares can be made.

19. Essential Understanding
Mathematical Essential Understanding Halves Take Up the Same Amount of Space
3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Directions:
As participants are working, ask probing questions. The goal is for participants to identify an essential understanding related to the objective on the chart.
Possible Probing Questions and Possible Responses:
If students know that 𝟑 𝟒 on the right and 𝟏 𝟐 are equal, then what must be true?
They each take up the same amount of area.
It doesn’t look as if they take up the same amount of area. What would a student have to do to prove this?
Cut out the 𝟑 𝟒 and lay it on top of the half.
Objective
Students recognize half of a figure as two spaces of equal size.
Essential Understanding

20. Essential Understanding
Mathematical Essential Understanding Continuous and Discrete Figures Represent a Whole
3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Directions:
As participants are working, ask probing questions. The goal is for participants to identify an essential understanding similar to the one on the chart.
Possible Probing Questions and Possible Responses:
How can both figures show halves?
There are three in the one to the right. It seems as if this one does not show halves.
What must be true for a student to show he or she understands that the figure on the right shows halves?
It shows halves. They are just spread out.
How do you know it is still half? How can you prove it?
Move the pieces around so you have all of the shaded together and all of the white pieces together. Put the shaded on top of the unshaded to see if there is the same amount.
A figure still shows halves even if the pieces are not located in one area. As long as the amount of space taken up by the figure is equal to the amount of space not shaded, then the figure shows halves.
Objective
Students recognize equivalent fractions as those that have the same amount of space of a figure.
Essential Understanding

21. Essential Understandings
CCSS
Equal Size Pieces
A fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.
3.NF.A.3d
Half of the Denominator is Half of the Whole
The denominator (bottom number) tells how many equal parts into which the whole or unit is divided. The numerator (top number) tells how many equal parts of that subdivided whole are indicated. 3 equal pieces is half of 6 pieces because makes 6. 2 equal pieces is half of 4 pieces because makes 4.
3.NF.A.1
Halves Take Up the Same Amount of Space
A fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but regardless of the model what remains true about all of the models is that they represent equal parts of a whole.
3.NF.A.3a
Continuous and Discrete Figures Can Both Represent Halves
Figures show halves if the same amount of area exists for each half, regardless of the location of the areas in the figures.
3.NF.A.3b
Not all Halves are Created Equal
A Whole Can be Represented as a Fraction
If all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented.
2 2 = = = 1
3.NF.A.3c
Directions:
Read the essential understandings. Do you agree with how they are aligned with the standards? Why or why not?
Probing Facilitator Questions:
What is meant by continuous and discrete figures? Point to a figure that shows halves in a continuous way. Point to a figure that show halve illustrated in a discrete way. Which is harder for students and why?
Why does 𝟐 𝟐 , 𝟑 𝟑 , and 𝟒 𝟒 show 1? What is true about all of these? Is this important for students to know?
If students know that half of the denominator is half of the whole, then what questions might you ask to see if they understand this idea?
The EU about half of the denominator seems different from the EU about the same amount of space. Can you say how the two ideas are talking about different ways of talking about half?

22. Students’ Mathematical Understandings
Assess ?
Target
Mathematical Goal
Directions:
(SAY) We start with a group of students in our class. It is not clear what they understand. There are a lot of different understandings, but they must be uncovered. Assessing questions do this.
What do the colors represent?
Why are the circles placed randomly?
Students’ Mathematical Understandings

23. A Student’s Current Understanding
Advance ?
Mathematical
Trajectory
A Student’s Current Understanding
Target
Mathematical Goal
Directions:
(SAY) The purpose of the lesson is to help students move towards the mathematical goal. This implies that we MUST identify our goal prior to beginning the lesson.
An advancing question must take into consideration both the student’s current understanding and the mathematical goal we are moving towards. It is the question that facilitates the student’s movement towards the goal, as s/he grapples with it. Note the trajectory takes the student from where s/he is towards the goal.
Is it just any question? I want you to consider what kind of question is needed to allow the student to get on the skateboard and move forward.

24. Mathematical Understanding
Target
Mathematical Understanding
Directions:
(SAY) In reality we have a classroom of students starting from different points, all aiming for the same goal. Although they are starting from different points, what do you notice about the students in your class?
Some have similar thinking.The yellow students think alike, the green students think alike, and so forth. So you really do not have to think about 30 students. You are really thinking about 5 or 6 students with the same kind of thinking. Thus our assessing and advancing questions will be different for different students.
Illuminating Students’ Mathematical Understandings

25. Characteristics of Questions that Support Students’ Exploration
Assessing Questions
Based closely on the work the student has produced.
Clarify what the student has done and what the student understands about what s/he has done.
Provide information to the teacher about what the student understands.
Advancing Questions
Use what students have produced as a basis for making progress toward the target goal.
Move students beyond their current thinking by pressing students to extend what they know to a new situation.
Press students to think about something they are not currently thinking about.
Directions:
(SAY)
On display is a list of the characteristics of assessing and advancing questions. Take a minute and read these. If you attended the session last summer, you will recall generating this list of characteristics in the past. If not, then note the characteristic of each kind of question.
Probing Facilitator Question and Possible Responses:
How are assessing and advancing questions related to the Standards for Mathematical Content and the essential understandings?
It is our responsibility to move students toward the standards. The EUs are what we want to hear if they are moving toward the standard.
Now we are going to write assessing and advancing questions again. This time we are going to keep in mind the essential understandings.
If the article, “Classroom Assessment: Minute by Minute, Day by Day,” has been distributed and read by the group, you might consider making connections to the article at this time.
Assessing and advancing questions are essential in achieving quality assurance in the classroom. In the article, “Classroom Assessment: Minute by Minute, Day by Day,” quality assurance involves a shift of attention from teaching to learning. The article suggests teachers must ask students “questions that either prompt students to think or provide teachers with information that they can use to adjust instruction to meet learning needs.” The article suggests “teachers listen evaluatively rather than interpretively.”
Which type of question is probably easier to craft? Why?

26. Essential Understandings
CCSS
Equal Size Pieces
A fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.
3.NF.A.3d
Half of the Denominator is Half of the Whole
The denominator (bottom number) tells how many equal parts into which the whole or unit is divided. The numerator (top number) tells how many equal parts of that subdivided whole are indicated. 3 equal pieces is half of 6 pieces because makes 6. 2 equal pieces is half of 4 pieces because makes 4.
3.NF.A.1
Halves Take Up the Same Amount of Space
A fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but regardless of the model what remains true about all of the models is that they represent equal parts of a whole.
3.NF.A.3a
Continuous and Discrete Figures Can Both Represent Halves
Figures show halves if the same amount of area exists for each half, regardless of the location of the areas in the figures.
3.NF.A.3b
Not all Halves are Created Equal
A Whole Can be Represented as a Fraction
If all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented.
2 2 = = = 1
3.NF.A.3c
Directions:
Assign each group one of the student work samples A through E.
(SAY) Study the student’s work sample. Determine the essential understanding(s) the student knows and the essential understanding(s) that needs to be developed. Write two assessing and two advancing questions that might be asked. Be prepared to explain what you hope to achieve/hear when you ask the advancing question. We will come back together as the larger group and share our work.
Facilitator Questions:
What have the students done?
How do you know? What on the paper tells you this?
Watch for inferences. If an inference is made, then say, “Do we know that or would we have to ask the student this question?”

27. Supporting Students’ Exploration Analyzing Student Work
Analyze the students’ responses.
Analyze the group work to determine where there is evidence of student understanding.
What advancing questions would you ask the students to further their understanding of an EU?
Directions:
Read the directions.

28. Group A: Lauren and Austin
Facilitator Questions:
What does the student know? What is unclear in the student work?
Assessing:
Tell me more about this first one. You say that it is half because each part is the same size. I see four pieces shaded though so how can this be halves? OR
Tell me about the second figure.
Advancing:
Can you write a fraction that tells me about their first picture? Then write a fraction for the last picture also.
You said the first figure shows halves and the last one has two shaded and two not shaded. Do both of these figures show halves? How can you prove this?

29. Group B: Jacquelyn, Alex, and Ethan
Facilitator Questions:
Assessing:
Tell me about the sixths. How do you know how many make halves?
Advancing:
If you look at 8ths then what fraction is needed to make halves? ( 4 8 ) How do you know this without even looking at it? Be prepared to explain. If you look at 10ths what would half be?

30. Group C: Tylor, Jessica, and Tim
Facilitator Questions:
Assessing:
How do you know that the first picture on the left shows halves?
Advancing:
If “a” is halves, then find another one that shows halves. How can the second to the right show halves? How can you prove that this is cut into halves?

31. Group D: Frank, Juan, and Kimberly
Directions:
Assessing:
Tell me about the two that you say are the same. OR
Does the figure in the middle show halves? How can you prove that it shows halves?
Advancing:
What fractions can you write to describe each of these figures? Can you think of two different fractions for each?
Let’s look at this figure. It has six pieces. Does it show halves?

32. Group E: JT, Fiona, and Keisha
Facilitator Questions:
Assessing:
Tell me about this figure that you circled. Does it show halves? I see you wrote four. Can you say more?
Advancing:
Prove that this figure on the right shows halves. I don’t believe it shows halves, convince me it does.

33. Reflecting on the Use of Essential Understandings
How does knowing the essential understandings help you in writing advancing questions?
Directions:
(SAY) Does it really matter? Does it really help to have EUs when you go into the lesson? Aren’t the standards good enough?
Probing Facilitator Question and Possible Responses:
How does knowing the essential understandings help you in writing advancing questions?
I clearly know what I am targeting with my questions. The EUs tell me what students must understand.
How do the standards differ from the EUs?
The standards are more global. They do not tell you what mathematical ideas