mathematics performance tasks · 2020-02-14 · mathematics performance tasks! 1"...

Mathematics Performance Tasks

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CREATING AND SOLVING WORD PROBLEMS

Overview

At a Glance In this activity, students observe a situation in a picture, make comments about whatever they notice, and then pose mathematical questions about the situation. Then individuals answer the questions posed by the class.

Grade Level Grade 1

Task Format • Small group or whole class and partner work • Three parts of increasing challenge to be used when students are

ready for each part (i.e., they do not need to be done on consecutive days)

Materials Needed For each student • 1 pencil • a writing surface, such as a table or clipboard • 3 copies of the Problem Solving Template (provided) • counters, cubes, buttons, or base-‐10 blocks, if needed by students For the teacher • Part 1: 1 large copy of Picture A: Strawberries and Cherries; counters,

cubes, or buttons • Part 2: 1 large copy of Picture B: 9 cupcakes; 1 large copy of

Picture C: 7 cupcakes; counters, cubes, or buttons • Part 3: story visible on a board or chart paper; counters, cubes,

buttons, or base-‐10 blocks (1 ten and 10 ones) • Observation Checklist

Prerequisite Concepts/Skills

• Counting to 100 by ones and by tens • Reading and writing numbers 0–20 • Experience in representing addition and subtraction situations with

objects, fingers, drawings, or acting out situations • Familiarity with combining collections to find totals 5 ≤ n ≤ 10 • Experience with composing and decomposing n ≤ 5 into pairs

Content Standards Addressed in This Task

1.OA.A.1

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

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Extensions and Elaborations

1.OA.A.2

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects drawings, and equations with a symbol for the unknown number to represent the problem.

Standards for Mathematical Practice Embedded in This Task

MP1 Make sense of problems and persevere in solving them.

MP4 Model with mathematics.

GET READY: Familiarize Yourself with the Mathematics This task assesses students’ understanding of how to solve various addition and subtraction situations involving totals ≤ 20. The task is presented in three parts, broken into types of addition and subtraction situations (detailed below) of increasing difficulty.

Part 1: Students work with addition and subtraction situations in which all numbers are ≤10. Part 1 focuses on problems in which the result is known, but one of the addends is not. As you use this activity with your students, you might find that some need review of the Result Unknown type problems, which are a major focus of kindergarten instruction.

Part 2: In this part, students are working with Compare type problems in which the difference between two quantities is the focus. More information is detailed below, but these are typically more difficult for students.

Part 3: Students solve problems similar in type to Parts 1 and 2, but now with numbers ≤ 20.

Each part is its own task. You may choose to begin all students on Part 1 and, depending upon your observations, move some students to Part 2 and Part 3; or you may implement Parts 2 and 3 at a later date, entirely at your discretion. Also, depending on your mathematics program, you might do Part 3 before Part 2.

Question-‐less Word Problems Traditional word problems typically end with a question that students need to answer. In this task, students look at a picture and describe what they see. Then they come up with the question(s) to be answered. This type of presentation engages students and gives them an entry point to the problem—any observation is valid. It also gives you, the teacher, insight into the type of thinking coming from your students. They might suggest many non-‐mathematical observations or even stories about their experience with the objects portrayed. As the activity continues, you will want to move them to the mathematical observations—quantities, categories, patterns, etc.—by sorting the students’ comments into two lists. From here, it might be a natural transition, or it might need some prompting, to move students to posing the mathematical questions that can be answered through the information in the picture. This gives them ownership of the solution process and the result. When they are outside of the classroom, they will begin to see the world around them through a more mathematical eye. Beyond engagement, this presentation method gives students experience with making sense of a situation and helps them learn to distinguish important information from the irrelevant.

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Types of Addition and Subtraction Situations Problem solving in grade 1 helps students learn to mathematize and model addition and subtraction situations using objects, drawings, and equations. This becomes a foundation for later algebraic problem solving (NCTM, 2009). When presented with a story problem, students must first make meaning of the situation. From this understanding, they can build a representation of the problem that helps them choose or develop a strategy for finding a solution. The representation may take the form of objects, drawings, or written equations. Standard 1.OA.A.1 calls for students to become proficient in solving three different types of addition and subtraction situations: Add To/Take From, Put Together/Take Apart, and Compare. In grade 1, each of these situations involves three quantities and students need experience with problems that have unknowns in all positions. For each of these, students should use the representation that best suits where they are: one student may be ready to solve a problem with a written equation; another student may be just as capable of solving the problem, but does it using objects for representation. Variations among students are certainly expected and the type of representation used is worth noting. Finally, keep in mind that grade-‐1 students should be able to represent these situations using equations, so it is important to help students move towards that goal. Add To/Take From. Standard 1.OA.A.1 uses the term Add To/Take From for situations that have also been called Change Plus/Change Minus (Cross, Woods, & Schweingruber, 2009; NCTM, 2009). Such problems involve three quantities A + B = C or A – B = C, in which A is the starting quantity, B is the amount by which this quantity changes, and C is the result. This problem type is probably most familiar:

Miriam has 7 crayons and Joshua gives her 4 more. How many crayons does Miriam have now? or

Miriam had 7 crayons but gave 4 to Joshua. How many crayons does Miriam have now?

It is called Result Unknown, because A and B, the start and the change, are known and the student must determine C, the resulting sum or difference. This is how the facts are typically taught—e.g., 7 + 4 = ☐, or 7 – 4 = ☐. Because of this, A + B = ☐ or A – B = ☐ become the most familiar and, generally, the easiest forms for children to solve. Reversing the action results in Start Unknown ☐ + B = C or Change Unknown A + ☐ = C. More experience with these increases children’s familiarity and proficiency with them. Table 1, from the Common Core State Standards for Mathematics (NGA & CCSSO, 2010, p. 88), gives examples of all these types. Because standard 1.OA.A.1 calls for unknowns in all positions and because kindergarten students solved situation types with an unknown result and unknown total, your grade-‐1 students may be ready to focus more on Change Unknown and Start Unknown situations.

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Table 1. “Add To” and “Take From” Story Problem Type Result Unknown Change Unknown Start Unknown

Add to

A bunnies sat on the grass. B more bunnies hopped there. How many bunnies are on the grass now?

A + B = !

(e.g., Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = !)

A bunnies were sitting on the grass. Some more bunnies hopped there. Then there were C bunnies. How many bunnies hopped over to the A bunnies?

A + ! = C (e.g., Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ! = 5)

Some bunnies were sitting on the grass. B more bunnies hopped there. Then there were C bunnies. How many bunnies were on the grass before?

! + B = C (e.g., Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ! + 3 = 5)

Take from

C apples were on the table. I ate B apples. How many apples are on the table now?

C – B = ! (e.g., Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = !)

C apples were on the table. I ate some apples. Then there were A apples. How many apples did I eat?

C – ! = A (e.g., Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ! = 3)

Some apples were on the table. I ate B apples. Then there were A apples. How many apples were on the table before?

! – B = A (e.g., Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ! – 2 = 3)

Put Together/Take Apart. In Put Together situations, there are two separate parts (A) and (B)—e.g., red and green apples—which, combined, make a total amount (C) (Cross, Woods, & Schweingruber, 2009). Again, students may be solving to find any of these numbers. However, in all cases, students are using the addition equation A + B = C as the foundation and making appropriate changes to the equation according to the context of the word problem (e.g., A + ☐ = C; ☐ + B = C; or C – ☐ = B or A). Take-‐Apart situations work in the reverse. In these problems, the total (C) is known, but one or both of the parts (A) or (B) are not. Here students determine an unknown part or find all of the ways to break the sum into two parts when both are unknown. Table 2 below (NGA & CCSSO, 2010, p. 88) shows examples.

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Table 2. “Put Together” and “Take Apart” Story Problem Type

Total Unknown Addend Unknown Both Addends Unknown

Put Together/ Take Apart

A red apples and B green apples are on the table. How many apples are on the table?

A + B = ! (e.g., Three red apples and 2 green apples are on the table. How many apples are on the table? 3 + 2 = !)

C apples are on the table. A are red and the rest are green. How many apples are green?

A + ! = C C – A = !

(e.g., Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ! = 5 or 5 – 3 = !)

Grandma has C flowers. How many can she put in her red vase and how many in her blue vase?

! + ! = C (e.g., Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?)

5 = 0 + 5, 5 = 5 + 0

5 = 1 + 4, 5 = 4 + 1

5 = 2 + 3, 5 = 3 + 2

Compare. These situations involve finding the exact amount by which two quantities differ. Students may see this difference as either the “extra leftovers in the bigger quantity or the amount the smaller quantity needs to gain to be the same as the bigger quantity” (NCTM, 2009, p. 41). Table 3 below (NGA & CCSSO, 2010, p. 88) shows examples of these three sub-‐types. Table 3. “Compare” Story Problem Type

Difference Unknown Bigger Unknown Smaller Unknown

Compare

“How many more?” version: Lucy has A apples. Julie has C apples. How many more apples does Julie have than Lucy?

A + ! = C (e.g., Lucy has 2 apples. Julie has 5 apples. How many more apples does Julie have than Lucy? 2 + ! = 5)

“More” version: Julie has B more apples than Lucy. Lucy has A apples. How many apples does Julie have?

A + B = ! (e.g., Julie has 3 more apples than Lucy. Lucy has 2 apples. How many apples does Julie have? 2 + 3 = !)

“Fewer” version: Lucy has B fewer apples than Julie. Julie has C apples. How many apples does Lucy have?

C -‐ B = ! (e.g., Lucy has 3 fewer apples than Julie. Julie has 5 apples. How many apples does Lucy have? 5 – 3 = ! )

“How many fewer?” version: Lucy has A apples. Julie has C apples. How many fewer apples does Lucy have than Julie?

C – A = ! (e.g., Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 5 – 2 = !)

“Fewer” version with misleading language: Lucy has B fewer apples than Julie. Lucy has A apples. How many apples does Julie have?

B + A = ! (e.g., Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 3 + 2 = !)

“More” version with misleading language: Julie has B more apples than Lucy. Julie has C apples. How many apples does Lucy have?

! + B = C (e.g., Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? ! + 3 = 5)

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In the Difference Unknown situation, a student may think of Lucy’s and Julie’s apples using this type of representation:

A student may then use either addition or subtraction correctly to solve this problem. For example, a student may think, “I know that Julie has 5 apples, so if I subtract Lucy’s 2 apples, I will know how many more apples Julie has than Lucy (or how many fewer apples Lucy has than Julie).” Or, a student may think, “If Lucy has 2 apples and Julie has 5 apples, then 2 + ! = 5.” Either of these ways of thinking makes sense given the situation. Generally, Compare situations have been treated as if only a subtraction equation was legitimate to use. This understanding of Compare situations is too narrow. Remain aware of this as you observe your students, because students think about these situations differently.

Standards for Mathematical Practice The main purpose of word problems in a mathematics classroom is to prepare students for making sense and solving problems that arise in later classes or outside of the classroom. Even young students need opportunities to gather information, make sense of a problem, consider various approaches, be flexible and have stamina, and solve and check the reasonableness of their thinking. This is the essence of the Standards for Mathematical Practice, especially MP1. In school, word problems are often the mechanism to give students this experience, but recall that word problems are not a goal in themselves; their real purpose is to let students mathematize contexts that arise. This task exercises and builds habits of mind underlying two Standards for Mathematical Practice: MP1: Make sense of problems and persevere in solving them and MP4: Model with mathematics. Students get many chances to think about a situation, describe “the meaning of a problem,” and “look for entry points” to its solution. In doing so, students ask themselves, “Does this make sense?” given the context. They evaluate whether their approaches and solution are reasonable. Through these experiences, students develop an “I-‐can-‐puzzle-‐it-‐out disposition” (Goldenberg, 2015). This is all key to MP1. Students engage in MP4 as they apply the mathematics they know to solving problems that arise at home, with friends, or on the playground. In early grades, this might be as simple as writing an addition or subtraction equation to describe a situation, as in this task. Mathematically proficient students routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

For More Information Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds). (2009). Committee on Earth Childhood

Mathematics, Center for Education, Division of Behavioral and Social Science and Education & National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press.

Goldenberg, E. P. Mark, J., Kang, J., Fries, M., Carter, C., and T. Cordner. (2015). Making sense of algebra: Developing students’ mathematical habits of mind. Portsmouth, NH: Heinemann.

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2

Julie

Lucy

Difference Unknown

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National Council of Teachers of Mathematics (NCTM). (2009). Focus in grade 1: Teaching with curriculum focal points. Reston, VA: Author.

Richardson, K. (2012). How children learn number concepts: A guide to the critical learning phases. Bellingham, WA: Math Perspectives Teacher Development Center.

GET SET: Prepare to Introduce the Task 1. Gather the materials listed on page 1. For all parts of the task, make 3 copies of the Problem Solving

Template for each student. Have counters, cubes, or base-‐10 blocks available for students who might benefit from them.

2. Model the activity to the whole class or a small group. Pair students ahead of time and have them sit together. They will need a writing surface, such as a table or clipboard. The GO section “Observations of Students” column may help you as you observe students’ work.

Introducing the Task The introduction to each of the three parts is similar though the picture or story changes. The basic structure of the introduction is described below. Throughout this document, when specific language is suggested, it is shown in italics.

1. I’ll show you a picture (or read you a short story – Part 3).

2. Take a look at the picture (or think about the short story). What do you see? (What is possible? Part 3)

3. Have students volunteer to model the situation.

4. Once the context of the story problem has been discussed, say

What questions can you ask about the picture (or short story)? Keep a record of the questions and observations, writing for all the students to see.

5. Now that we’ve thought of some possible questions, let’s choose one to answer.

6. Now, work on your own to solve the problem. Solve it any way you choose—using objects, making drawings, or writing an equation. Use your paper to show how you solved the problem.

7. When you finish, talk with your partner about the strategies you used. Take turns so each of you has a chance to explain your thinking and your solution and also explain why you agree or disagree with your partner’s thinking.

8. If students are ready for it, you could also ask them to pose other questions and give those to a partner to solve.

Preparing to Gather Observation Data and Determine Next Steps in Instruction As students engage in the task, the notes in the next section will help you identify students’ current strengths and possible next steps for instruction. As you observe, use whichever form of the Observation Checklist that best helps you record your observations of students and other relevant evidence as you see it: Individual, Partner, or Class. These varied forms, available at the end of this document and in a separate MS Excel file, are intended to give you a choice about how to collect notes on your students and determine possible next steps for instruction.

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Addressing Student Misconceptions/Errors As students work, you may observe these common challenges: • Reading comprehension is a major factor in understanding both the context of a word problem and

the question it poses. Some of your students are still developing their comprehension strategies as well as their mathematical vocabulary. Take notice of words, phrases, or problem situations that appear to be particularly challenging for your students.

• Keep in mind that the language used to ask questions in word problems can vary but the meaning may be the same. For example, “Maria has 9 red marbles in her jar. Joshua has some white marbles in his jar. Together, they have 15 marbles.” Two questions that could be asked “How many white marbles does Joshua have in his jar?” OR “How many marbles would Joshua need to have in his jar so that they have 15 marbles in the two jars?” These two questions have the same answer and require the same arithmetic but are very different in syntax and sentence structure and even meaning. This may seem inconsequential to adults but can be confusing to students. Pay attention to how your students respond. Are there specific words, phrases, or questions that a student understands with ease? Are there others that are more challenging? Identifying the answers to these questions will help you uncover misconceptions or misunderstandings of language.

• When students’ answers are incorrect, try to discover why. Was the incorrect total caused by a counting error? A student’s misunderstanding of the operation of addition or subtraction? A developing understanding of language? It is valuable to pinpoint why this error took place. Asking students to explain their work can help you know which strategies students used and how they used them.

• Comparison situations in which the word in the story is the opposite of the action needed for the solution are confusing for students (NCTM, 2009). For example, let’s use the following problems to illustrate this issue:

Jocelyn has 8 pears. She has 3 fewer pears than Juan. How many pears does Juan have?

Jacob has 8 apples. He has 3 more apples than Jenny. How many apples does Jenny have?

For many students, the word “fewer,” if it is understood at all, screams to subtract, and the word “more” suggests they should add. However, to know how many pears Juan has, they must add 3 to 8, not subtract 3 as students may think; and to know about Jenny’s apples, they must subtract. Although this kind of Compare situation is more common in grade 2, you may have students who are ready to give it a try. If so, be mindful of these types of comparison situations and take notice of how your students interpret the language.

Extensions and Elaborations This task can be extended in a variety of ways: • You might invite students to ask other questions about a particular picture (from Parts 1 or 2) or

scenario (Part 3), to choose one question, and then solve that question. For example, other questions that students may be interested in solving include: − “How many sprinkles are there on all the cupcakes?”

− “How many more cherries does ______ [name of Student A] have than _______ [name of Student B]?”

• You might use other pictures or other question-‐less word problems for which students can create a question, just as they have in this task. To do this, you will need a picture that is rich enough for

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students to identify various quantities and ask questions. For a question-‐less word problem, you can often find a word problem that you have used in the past and remove the question.

• To address standard 1.OA.A.2, which calls for students to solve word problems that involve adding three whole numbers whose sum is ≤ 20, you may decide to provide students with other question-‐less word problems that include three addends. For example: “Julie has 8 stickers. Diego has 4 stickers. Sandy has 5 stickers.” You can follow this with “What questions can you ask?” This will prompt students to come up with questions (e.g., “How many stickers do they have in all?” or “If they shared as evenly as possible, how many would each have?”), which they will then solve. Of course, not all of the questions will be about adding: “Who has the most?” is a very sensible mathematical question even though it does not address this particular standard.

• Another option is to have each player explain his or her partner’s solution. In doing so, each partner will have to have a deep understanding of the other’s explanation and strategy to be able to explain clearly. Each set of partners can then pair up with another set to share their thinking.

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GO: Carry Out the Task Part 1: Add To/Take From – Change/Start Unknown or Put Together/Take Apart—Addend Unknown word problems with totals ≤ 10

Task Steps Keep in Mind Observations of Students

1. Show either a small group of students or your whole class Picture A. Ask students to describe what they see. Accept any descriptions that the students provide.

SAY to STUDENTS:

Tell me what you see in this picture.

Yes, and what else can you say?

Try to elicit as many observations as the students are willing to provide. Maybe include one of your own.

Avoid asking leading questions like “How many ___ do you see?” or “Are they all the same size?” Let students add those details spontaneously as they try to find more to say about the picture. Such added detail builds toward proficiency in MP6: Attend to precision.

• This task can give you insights into your students’ thinking: − What do students observe? − Do students count, add, subtract? − Do students spontaneously comment

on size or quantities (e.g., large and small strawberries, pairs of 2 cherries, the number of fruits by category or total)?

• The design of the art is intentional: multiple sizes of strawberries with different numbers of seeds and sets of cherries. These variations create the possibility for a number of observations and numerical combinations.

• At this point in the task, if students attach calculations to their descriptions (e.g., saying “I see 3 small strawberries, 2 large strawberries and 8 cherries, so that’s 13 pieces of fruit”), accept it, but don’t push for it. The goal in this portion of the task is descriptions of the picture, attending to increasing detail; other sections will address students’ ability to attach and perform calculations.

A. Student gives a single qualitative description, like “fruit” or “strawberries,” with no further detail, such as number or size.

B. Student categorizes by only one attribute (fruit type or size) and does not include number as part of the description (e.g., student says “strawberries and cherries” or “large fruits and small fruits”).

C. Student includes a single quantitative description, for example, counting all objects together or counting only one subset (by size or kind), but does not count more than one subset (e.g., student says “I see five strawberries,” but does not mention or count the cherries).

D. Student is able to categorize in several different ways (type of fruit, size, etc.) and names the quantities of at least some of these sets.

2. Next, call up two students. Using no • Why use invisible objects? One reason is E. Student benefits from using physical

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objects of any kind, pretend to put 3 strawberries in Student A’s hand.

SAY to ALL STUDENTS:

I have 7 invisible strawberries in my hand! I’m going to give _____ [student’s name] 3 strawberries.

Nothing is there, of course, but playfully ask the student to “check” to make sure that the right amount is “there.” If the student does not seem to realize that the whole game is pretend, make that clear.

Then move to Student B. Pretend to put a secret number of strawberries in his or her hand.


Now I’m going to give the rest of my strawberries to _____ [student’s name].

Note: While this step uses “invisible objects,” you should feel free to model this with physical objects (e.g., counters, buttons) if you know that this will benefit some of your students.

that mathematics depends on many foundations, and one of them is good working memory. Students expand their capacity when they mentally visualize quantities and hold multiple pieces of information in their heads. By not providing counters or a picture, we also move students towards mental computation instead of counting to come up with the response. This formative assessment task gives insight into students’ progress in this area.

• Why go through the extra step of having the student “check” things that don’t exist? Students generally find this step funny and a nice invitation to dive into the problem with all their ability to pretend. But there’s also a serious mathematical side. Asking the student to “check” focuses attention on the number, because the child has to pretend-‐count, and also helps the child create a mental image of the objects in each hand.

objects (e.g., counters or cubes) in place of “invisible objects.”

F. Student is able to visualize the number of “invisible objects” by telling the correct number, when prompted.

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3. Now that you have set the context,

SAY to STUDENTS:

What questions can you make up about this situation?

Depending on the prior experiences of the students in your class, you might word this question differently. Other options include:

• If you were going to make up a story problem, what questions might you include in your problem?

• So far, the story problem says “_____ [Student A] has 3 strawberries. _____ [Student B] has some strawberries. What questions can you ask to make this into a word problem?”

Accept all questions that your students generate, whether or not they are mathematical. Feel free to point out which are and are not mathematical but don’t have your students restrict their responses at this point. If you get only one question, prompt for others.

SAY to STUDENTS:

Can anyone think of any more questions?

At this age, and especially the first time, children may well run out of questions after only two or three. Count in your head

• Depending on the time of year, grade-‐1 students may not reliably distinguish between questions and observations. Moreover, they may not differentiate between those that are mathematical (more, less, total) from ones that are not.

• The mathematical questions that students ask may be restricted to types they have become familiar with in class. You may want to add one less-‐conventional question of your own to expand their repertoire.

• While many questions are possible, some of the following are quite common and others are rare. − How many strawberries do Students A

and B have in all? − Who has more strawberries? − How many more strawberries does ___

[Student B] have? − How many fewer strawberries does ___

[Student A] have? − Student A wants to have as many as

Student B. How many more does he or she need?

− I ate strawberries today for snack! − If _____ [Student B] takes one of ____

(Student A’s] strawberries, how many does he or she have now?

− If they combined them, and then shared them equally, how many would each have?

G. Student makes an observation rather than asking a question.

H. Student requires support (teacher or peer) to generate a question.

I. Student asks a relevant question, but only about the stated facts of the situation (“How many strawberries does ____ have?”), not about unstated information that can be derived from those facts (such as who has more).

J. Student spontaneously offers more than one relevant mathematical question.

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to 20 to leave wait-‐time for thinking, but if nothing comes,

SAY to STUDENTS:

OK, well maybe there aren’t any more good ones!

4. SAY to STUDENTS (for example, to the students with invisible strawberries in their hands):

Now, let’s solve a story problem. Remember _____ [Student A] has 3 strawberries and _____ [Student B] has the rest of the 7 I started with. How many does [Student B] have?

Solve this problem any way you choose—using objects, making drawings, or writing an equation. Use your Problem Solving Template to show what you did. At the end, you and your partner will talk about your strategies.

5. Allow students a few minutes to work individually to solve the problem and show their work on the Problem Solving Template. Students may demonstrate their thinking by using physical objects, making drawings to represent the problem, or writing an equation.

Note: When students use objects, you may want to take a picture of the

• Do students use physical objects (e.g., counters or base-‐10 blocks) to represent the situation? If so, how do students use them?

• Do students create a drawing to represent the situation? If so, does the drawing accurately match the context of the story? (The drawing does not need to look like strawberries. It can accurately match the context with abstractions, like tick marks or dots or squares.)

• Do students approach the problem by writing an equation? If so, do the equations correctly match the situation?

• Do students use a combination of physical objects, drawings, or equations?

• Do students label the drawing or total? • Do students correctly represent one aspect

of the problem but make an error in finding the total? If so, what is the cause of the error?

K. Student uses physical objects to represent and solve the problem.

L. Student makes drawings to represent and solve the problem.

M. Student writes equations to represent and solve the problem.

N. Student labels his or her drawing or total. O. Student shows some understanding of the

scenario, but makes an error in his or her solution.

P. Student calculates the solution to the problem correctly.

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representation as a record of the work.

6. When students have solved and recorded the problem on their own, ask them to share their solutions with their partners and ask both partners to decide whether they agree or disagree with their partner’s work and to explain their thinking.

• Some students will benefit from having access to sentence starters to provide language support to their explanation. It may be helpful to have some available to students, posted in a visible location (e.g., white board, sentence strip). For example:

“I solved my problem by…” • How do the students explain how they

solved their word problem? • Do students provide an explanation for

each step of the problem? • How do students respond to each other? • Do students correct any of their partner’s

errors? If so, do they explain their thinking in a clear manner, so that their partners understand the error?

Q. Student provides little to no explanation of the reasoning used to solve the problem, even with the support of a sentence starter.

R. Student attempts to explain the reasoning. However, the explanation is often incomplete or flawed.

S. Student is able to explain the reasoning. Student’s explanation is thorough and complete. Student requires no additional support (e.g., sentence starters) when responding.

7. Create an Add To/Take From—Start or Change Unknown situation using the same picture.

For example, put some imaginary cherries in a student’s hand.


_____ [name of student] has 7 cherries. But he [or she] got hungry and ate some of them. Have student eat some in secret, hiding the number eaten. He [or she] now has 2 cherries left. What questions can

See Steps 4–6 above.

See Steps 4–6 above

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you ask? Collect a few questions and then choose one or two for students to solve.

Let pairs of students solve the problem on their Recording Sheet, and then discuss their strategies with their partner.

8. Create a third problem that involves a Put Together/Take Apart—Both Addends Unknown using the same picture. Call up a student to model the situation. For example, using no objects of any kind, put 8 cherries (in bunches of 2) in the student’s hand. Then,


_____ (name of student) has 8 cherries. She wants to put some into two different containers. How many can she put in each container?

Let pairs of students solve this problem on their Recording Sheet and then discuss their strategies with their partners.

Students have completed Part 1 when they have had the opportunity to solve a variety of the problem-‐types—Add To, Take From, Put Together and Take Apart—with Start or Change Unknown and one or both of the Addends Unknown. Provide students with as many opportunities as necessary in solving these types of problems. Feel free to add Result Unknown or Total Unknown as well, but remember these are the focus of kindergarten. You may continue to Part 2 whenever you feel students are ready. (It need not be the next day.)

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Part 2: Compare Problems with totals ≤ 10

Part 2 is similar in structure to Part 1 of this activity (detailed in the previous pages), but the focus of the problems in this part are Compare problems, still with all numbers ≤ 10. The steps include presenting students with Picture B or C (illustrated here and provided separately in this document) and asking students to make observations and pose questions. The information about student understanding and strategies for solving problems is the same. The difference in this part is that the questions you ask the students to solve are of a different subtype. These subtypes, Compare, are typically more difficult for students to understand and solve. Some examples based on these pictures include:

• Picture B examples might include: How many more vanilla frosted cupcakes are there than chocolate frosted? If Charlie has 4 fewer cupcakes than are in this picture, how many does he have?

• Picture C examples might include: How many more cupcakes have strawberries than have cherries? How many more sprinkles are on one cupcake than the other? Jarron has 5 fewer cupcakes than Mikala who has 12 cupcakes. How many does Jarron have?

Of course, there are many more questions of all types that can be posed from these pictures (or you could choose one of your own pictures from a story book or other classroom resource or context). Provide students with as many opportunities as necessary in solving these types of problems.

Picture C Picture B

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Part 3: Compare Word Problems and other Unknown Addend subtypes with Totals ≤ 20


1. Begin Part 3 by presenting students with Story D – Problem 1. You’ll want to have this problem visible to students on a board or chart paper.

SAY to STUDENTS:

Today we’re going to read a story problem. You’ll notice that something important is missing from the story. Let’s read the story to find out…

Read the following story aloud to students:

Jessie has 5 marbles. Maya has more than 10 marbles, but fewer than 16.

Pause for a moment to let students think about the scenario. Then,

SAY to STUDENTS:

What is possible?

Accept student responses. Students will likely generate possible amounts of marbles that Maya may have—or also may jump to finding a total for Jessie’s and Maya’s marbles.

For example, a student may say, “It is possible for Maya to have 11 marbles.” Or, a student may say, “If Maya has 13 marbles and Jessie has 5, they would have 18 marbles in all.”

• By asking a question-‐less word problem, you are developing students’ skills at using both natural language and mathematical language to describe ideas and develop mathematical meaning from real-‐world situations.

• Are students able to generate possibilities? If so, how do students go about it?

A. Student provides at least one possibility for the number of Maya’s marbles.

B. Student provides more than one possibility for the number of Maya’s marbles.

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2. Once students have generated a list of possibilities for the number of marbles Maya may have, tell students that today they are going to solve a story problem based on a few of these responses.

SAY to STUDENTS:

Problem 1: Today we’re going to solve a story problem about Jessie’s and Maya’s marbles. We already know that Jessie has 5 marbles, right? Let’s suppose Maya has ___ marbles.

Note: Choose a number from 11 to 15 for the number of Maya’s marbles.

How many more marbles does Maya have than Jessie?

You should solve this problem in any way you choose—using objects, making drawings, or writing an equation. You’ll do this on your Recording Sheet. At the end, you will have the chance to talk to your partners about the strategies you used.

Allow students a few minutes to work individually to solve the word problem and show their work on their Problem Solving Template. Students may demonstrate their thinking by using physical objects, making drawings to represent the problem, or writing an equation.

• Do students use physical objects (e.g., counters or base-‐10 blocks) to represent the situation? If so, how do students use them?

• Do students create a drawing to represent the situation? If so, do the drawings accurately match the context of the story?

• Do students attempt to solve the problem by writing an equation? If so, does the equation correctly match the situation?

• Do students use a combination of physical objects, drawings, or equations to solve the problem?

• Do students label their drawings or totals? • Do students correctly represent one aspect

of the problem, but make an error in solving for the total? If so, what is the cause of the error?

C. Student uses physical objects to represent and solve the problem.

D. Student makes drawings to represent and solve the problem.

E. Student writes equations to represent and solve problem.

F. Student labels his or her drawing or total. G. Student shows some understanding of the


H. Student calculates the solution to the problem correctly.

3. When students have solved the problem • Some students will benefit from having I. Student provides little to no explanation

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on their own, ask them to share their solutions with their partners and discuss whether they agree or disagree with their partner’s solution and thinking.

access to sentence starters to provide language support to their explanation. It may be helpful to have the following available to students, posted in a visible location (e.g., white board, sentence strip):

“I solved my problem by…” • How do students explain how they solved

their word problem? • Do students provide an explanation for

each step of the problem? • How do students respond to the other

player? • Do students correct any of their partners’

errors? If so, do students explain their thinking in a clear manner, so that the other players understand their errors?

for the reasoning used to solve his word problem, even with the support of a sentence frame.

J. Student attempts to explain her reasoning and provide a justification for the rationale used. However, the justification is often incomplete or flawed.

K. Student is able to explain her reasoning and provide a justification for the rationale used. Student’s explanation is thorough and complete. Student requires no additional support (e.g., sentence starters) when responding.

4. Repeat Steps 1 to 3 by posing different scenarios modeled after Problem 1 using the various sub-‐types of Compare. Allow students the opportunity to first solve each independently and then turn-‐and-‐talk to a partner about their strategies used.

Problem 2: SAY to STUDENTS:

We already know that Maya has more than 10 marbles, but fewer than 16, right?.

Let’s suppose Jessie has 4 more marbles

See Steps 1–3 above. See Steps 1–3 above.

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than Maya.

How many marbles could Maya have?

How many marbles does Jessie have?

Problem 3:

We know that Maya has more than 10 marbles, but fewer than 16, right?

Let’s suppose Jessie has 6 fewer marbles than Maya. How many marbles could Maya have? How many marbles does Jessie have?

Part 3 is complete when students have solved at least 3 different problems. Provide students with as many opportunities as necessary in solving these types of problems.

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OBSERVATION CHECKLIST

ASSESSING STUDENT UNDERSTANDING: WORD PROBLEMS WITH TOTALS ≤ 10 – PARTS 1 & 2 Use this page to record individual student observations. Use the letters to notate each event as you see it unfold. This record is intended to help you plan next steps in your instruction for your students.

Student Name Observations of Student Possible Individual Student Observations MATHEMATICAL OBSERVATIONS

A. Student gives a single qualitative description, like “fruit” or “strawberries,” with no further detail, such as number or size.

B. Student categorizes by only one attribute (fruit type or size) and does not include number as part of the description (e.g., student says “strawberries and cherries” or “large fruits and small fruits”).

C. Student includes a single quantitative description, for example, counting all objects together or counting only one subset (by size or kind), but does not count more than one subset (e.g., student says “I see five strawberries,” but does not mention or count the cherries).

D. Student is able to categorize in several different ways (type of fruit, size, etc.) and names the quantities of at least some of these sets.

MAKING MEANING E. Student benefits from using physical objects

(e.g., counters or cubes) in place of “invisible objects.”

F. Student is able to visualize the number of “invisible” objects by telling the correct number, when prompted.

MATHEMATICAL QUESTIONS G. Student makes an observation rather than

asking a question. H. Student requires support (teacher or peer)

to generate a question.

I. Student asks a relevant question, but only about the stated facts of the situation (“How many strawberries does ____ have?”), not about unstated information that can be derived from those facts (such as who has more).

J. Student spontaneously offers more than one relevant mathematical question.

REPRESENTATION K. Student uses physical objects to represent

and solve the problem. L. Student makes drawings to represent and

solve the problem. M. Student writes equations to represent and

solve the problem. N. Student labels his or her drawing or total. O. Student shows some understanding of the


P. Student calculates the solution to the problem correctly.

EXPLAINING REASONING Q. Student provides little to no explanation of

the reasoning used to solve the problem, even with the support of a sentence starter.

R. Student attempts to explain the reasoning. However, the explanation is often incomplete or flawed.

S. Student is able to explain the reasoning. Student’s explanation is thorough and complete. Student requires no additional support (e.g., sentence starters) when responding.

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OBSERVATION CHECKLIST

ASSESSING STUDENT UNDERSTANDING: WORD PROBLEMS WITH TOTALS ≤ 20 – PART 3 Use this page to record individual student observations. Use the letters to notate each event as you see it unfold. This record is intended to help you plan next steps in your instruction for your students.

Student Name Observations of Student Possible Individual Student Observations MAKING MEANING

A. Student provides at least one possibility for the number of Maya’s marbles.

B. Student provides more than one possibility for the number of Maya’s marbles.

REPRESENTATION C. Student uses physical objects to

represent and solve the problem. D. Student makes drawings to

represent and solve the problem. E. Student writes equations to

represent and solve problem. F. Student labels her drawing or

total. G. Student shows some

understanding of the scenario, but makes an error in her solution.

H. Student calculates the solution to the problem correctly.

•

EXPLAINING REASONING I. Student provides little to no

explanation for the reasoning used to solve his word problem, even with the support of a sentence frame.

J. Student attempts to explain her reasoning and provide a justification for the rationale used. However, the justification is often incomplete or flawed.

K. Student is able to explain her reasoning and provide a justification for the rationale used. Student’s explanation is thorough and complete. Student requires no additional support (e.g., sentence starters) when responding.

Creating and Solving Word Problems - Part 1: Picture A - Cherries and Strawberries

Creating and Solving Word Problems - Part 2: Picture B - Cupcakes 1

Creating and Solving Word Problems - Part 2: Picture C - Cupcakes 2

Creating and Solving Word Problems - Part 3: Story D - Problem 1

Jessie has 5 marbles.

Maya has more than (>) 10 marbles, but fewer than (<) 16.


Jessie has 4 more marbles than Maya.

How many marbles could Maya have?How many marbles does Jessie have?



Jessie has 6 fewer marbles than Maya.

How many marbles could Maya have?How many marbles does Jessie have?


My SolutionName Date

Solve the story problem in the space below.

Solve the story problem in the space below.

Creating and Solving Word Problems: Problem Solving Template

My Solution

Creating and Solving Word Problems: Problem Solving Template

Name Date

mathematics performance tasks · 2020-02-14 · mathematics performance tasks! 1"...

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