Unit 1.6: Problems with Unknowns

Addition and subtraction represent a relationship between quantities in a real world context. The context may represent an add to, take from, put together, take apart, or comparison situation which can be recorded symbolically with an equation. Any number in the relationship may be unknown.

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Unless otherwise noted, SFUSD Math Core Curriculum is licensed under the Creative Commons Attribution 4.0 International License

New Learning:

Re-engagement:

• In Unit 1.1, students worked with add to and put together situations where the sum is unknown to 10 and then the teen numbers.
• In Unit 1.3, students were exposed to add to and put together addition within 20, where the unknown is the sum or the change.
• In Unit 1.4, students connected unknown change addition problems to subtraction and worked with take from and take apart problem types.

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• Students represent and solve addition and subtraction word problems with the change or start unknown.
• Students understand the meaning of the equal sign through examining the relationship between equivalent expressions.
• Students create equivalent addition and subtraction equations.
• Students determine the unknown start or change quantity in addition or subtraction equations.
• Students represent and solve word problems that require comparison of quantities.

Problem Types

Add To and Take From – Start Unknown situations are more difficult than the Result Unknown and Change Unknown situations. First Grade introduces Start Unknown problem types. Mastery is expected in Grade 2. See the K-5 Progressions on Counting and Cardinality and Operations and Algebraic Thinking for more information on grade level expectations for these types of word problems.

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This unit concentrates on the Addition and Subtraction Situations highlighted in blue, in the following table, from the Glossary of the Common Core State Standards - Math. Helping students identify problem types can assist them in developing schemas, or frameworks, for interpreting word problems. As they gain familiarity with the problem types, they can begin to recognize new problems as having a familiar structure, and begin to use the same strategies and representations for similar problems. Using problem types to support students with understanding word problems has been shown to be more effective than using “keywords” to identify operations because keywords can often be misleading, while problem types may look different on the surface but have similar underlying structures.

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Problem types often can be solved using more than one operation depending on what information is known and unknown, so they are not categorized by a particular operation but rather by the relationships between the quantities. Tape diagrams can help students identify this underlying structure.

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Suggested Lesson Sequence: 10 Lessons to be taught over 20 days

January 4-15

Description:

Students solve addition and subtraction word problems and play a game where the change is unknown. Students represent the situations with tape diagrams and number lines.

Lesson 1: Entry Task Seesaw Lesson 1 - Pennies

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Lesson 2: LS 1 Day 2 - Seesaw Lesson 2 - Pennies and Nickels

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Lesson 3: LS 1 Day 3 - Seesaw Lesson 3 - Spinner Game

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Lesson 4: LS 1 Day 4 - Seesaw Lesson 4 - Spinner Game #2,

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Description: Students solve a start unknown problems in addition and subtraction. They start with a problem in context for both situations and then play a game for practice.

Lesson 5: LS 2 Day 1- Seesaw Lesson 5 - Pennies Start Unknown

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January 19 -29

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Lesson 6: LS 2 Day 2 Seesaw Lesson 6 - Start Unknown Game

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Lesson 7: LS 2 Day 3 Seesaw Lesson 7 - More Pennies

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Lesson 8: LS 2 Day 4 Seesaw Lesson 8 - Start Unknown Game #2

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Lesson 9: LS 3 Day 1 Seesaw Lesson 9 - Expert Task

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Milestone Task District Assessment Seesaw Milestone (Spanish), (Chinese)- Trisha’s Pennies

Synchronous and Asynchronous Teaching Options:

Use a combination of Synchronous and Asynchronous approaches

Additional Resource:

Splitting Screens on Zoom

You may be at a point in distance learning when you would like to have students work in another window while on Zoom if they don’t already. This allows them to do things like use virtual manipulatives to support their learning or work collaboratively in a Jamboard during live instruction.

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You may use these short instructional videos (one for Chromebooks, one for Macs/PCs) to teach students how to split their screens between Zoom and another window. You can:

• play the videos to students,
• record your own version, or
• explain how to do it live with students (best if done in small groups)

Chromebook

Mac/PC

Technology Resources

• Dots in a Box - Fluency practice for sums to 20

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• Georgia CCSS-M Standards videos is a website from the State of Georgia that includes videos of all the math content standards. For this unit, watch the video for 1.OA.7.

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• Virtual Math Balance here: http://www.didax.com/apps/math-balance/ Youtube tutorial here: https://www.youtube.com/watch?v=qIgX935xMPM

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Student PDFs

1.6 Classwork .S. .C.

1.6 Homework .S. .C.

Data Routine

Objective: The Data Routine has both social and a math objectives:

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Social

• Students build community by sharing about themselves - their person, lives, interests, & preferences - learning about their classmate’s, finding and celebrating their similarities and differences
• Students build community by recognizing their shared environment (e.g. weather)

Math

• Students develop their understanding of categorical and measurement data (see progression of standards in the following slides and here) as they:
• Formulate questions
• Collect data
• Organize and display the data
• Analyze the data, and
• Interpret the results
• Students apply other important math ideas from all domains

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See the Data Routine slides for a description of how to incorporate this routine into your lessons.

Daily Routine: Number of Days in School

Objective: To build one-to-one correspondence, lay the foundation for an understanding of place value, and give students a concrete sense of the magnitude of numbers up to 180 and their relationship to the passage of time.

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See the Number of Days in School (Spanish) slides for a description of how to incorporate this routine into your lessons.

Daily Routine: Counting Routine

Objective: To provide students opportunities to count forward and backwards by various whole numbers and decimals, developing an understanding of patterns in counting, addition and subtraction (and later multiplication) and place value.

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See the Counting Routine slides for a description of how to incorporate this routine into your lessons.

Math Talks:Re-engage with Number Lines

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Number Line Talk

Objective: Students develop deeper number sense and estimation skills by determining whether a number is closer to 0 or closer to 20 using an unmarked number line.

Description: These Math Talks are designed to encourage students to apply their knowledge of the number sequence, think about the relative distances between numbers, and strengthen their ability to anchor numbers to 0, 5, 10, and 20.

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Locating One Number

Question: Where should ___ be on the number line? Is __ closer to 0 or 10 or 20? How do you know?

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13

Where should ___ be on the number line? Is __ closer to 0 or 10 or 20?

How do you know?

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Math Talks:Engage with Current Content

True or False

Objective: To evaluate equations by applying various strategies to determine if they are true or false.

Description: These math talks are designed to encourage students to reason about numbers and operations, without necessarily solving the problems. Encourage students to look at the relationships between the expressions on either side of the equal sign to see if there is a reason the equation will be true or false before they find the solution. For example students may reason that 6 + 6 + 1 = 6 + 7 is true because 6 + 7 is double 6 plus 1, which could be written as 6 + 6 + 1. Use 1–2 equations per math talk at the most.

Suggested Math Talks:

Question: Is this equation true or false? How do you know? Can you think about it another way?

Is this equation true or false? How do you know?

Can you think about it another way?

8 + 3 = 8 + 2

12 - 4 = 11 - 3

20 - 2 = 18 + 2

Math Talk

Is this equation true or false? How do you know?

Can you think about it another way?

3 + 14 = 14 + 3

6 + 6 + 1 = 6 + 7

6 + 8 = 7 +7

Math Talk

Lesson 1: Entry Task

Whole Class or Groups: Launch- 3 - Read Protocol

• First Read: What is the story about?
• Second Read: What are the quantities and the units in the story?
• Third Read: What math question could we ask about the story?

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Independent work: Explore - Seesaw Lesson 1 - Pennies, Entry Task Pennies Student .S. .C. , Double ten frame, Number Lines 0 to 20 BLM

• Solve the Pennies problem - Use the tools or drawings to show your thinking.

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Whole Class or Groups: Summarize-

• Use examples provided or actual student work to discuss various strategies for solving problems with the change unknown
• Core Math to Emphasize: The change can be unknown in an add to situation. Change unknown situations can be solved by counting on, counting back or subtraction.

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Math Norms

18

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

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SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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Problems with Unknowns

1st Read

2nd Read

3rd Read

Pennies

Count On

Equation: 8 + ? = 13

Count Back

Equation: 13 - ? = 8

Subtraction

Equation: 13 - 8 = ?

Lesson 2: LS 1 Day 2

Whole Class or Groups: Launch- 3 - Read Protocol

• First Read: What is the story about?
• Second Read: What are the quantities and the units in the story?
• Third Read: What math question could we ask about the story?

Independent work: Explore - Seesaw Lesson 2 - Pennies and Nickels, Day 2 Pennies and Nickels Student .S. .C. Day 2 Pennies and Nickels Extension Problems BLM .S. .C.

• Have students help you fill in the tape diagram.
• Students use the tools to solve the Pennies and Nickels problems.

Whole Class or Groups: Summarize-.

• Share student strategies for solving
• Ask the students to come up with equations and list them under the tape diagram.
• Core Math to Emphasize: One addend can be unknown in a put together situation. Unknown addend situations can be solved by counting on, counting back or subtraction, and represented with a tape diagram.

Math Norms

28

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

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SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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1st Read

2nd Read

3rd Read

How many nickels does she have in her piggy bank?

Pennies and Nickels

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11

18 Coins

?

Pennies and Nickels

11

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18 Coins

?

Equations

Pennies and Nickels

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11

18 Coins

?

11 + ? = 18

18 – ? = 11

18 – 11 = ?

11

?

18 coins altogether

2

?

equations:

8 coins altogether

4

?

equations:

18 coins altogether

5

?

equations:

19 coins altogether

Lesson 3: LS 1 Day 3

Whole Class or Groups: Launch-

• Tell students today they are going to play a game where the unknown number is in the middle of the equation
• Spin the spinner for the first addend and pull a number card 6-20 for the result.
• Today they will use the number line to solve the problem. Allow students to use counters with their number lines as needed.

Independent work: Explore - Seesaw Lesson 3 - Spinner Game, Spinner , Addition Change Unknown Game Template BLM .S. .C., Number Lines 0 to 20 BLM

• Model a few more rounds by spinning the spinner or rolling a dice and choosing a number cards 6-20

Whole Class or Groups: Summarize- Choose 2–3 students who recorded the same problem and have them show how they solved it on the number line, or use these samples.

• Ask students if they thought of the same problem in a different way.
• Ask students to describe what the thinking is in each number line
• Ask students why each strategy works.

Core Math to Emphasize: Unknown change addition problems can be shown on a number line by counting on, counting back or subtraction.

Math Norms

42

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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Let’s Spin!

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6 + ? = 9

Spinner # Number Card

9

?

Let’s Spin!

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+ ? = 15

Spinner # Number Card

15

?

v

Let’s Spin!

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+ ? = _

Spinner # Number Card

?

v

Lesson 4: LS 1 Day 4

Whole Class or Groups: Launch-

• Remind students of the addition game they played yesterday with the unknown in the middle of the equation.
• Tell the students that today they are going to play the same game, but with subtraction.
• Ask students how they might use the number line to help them solve subtraction problems when they don’t know how much to take away.

Independent work: Explore - Seesaw Lesson 4 - Spinner Game #2, Spinner , Subtraction Change Unknown Game Template BLM .S. .C , Number Lines 0 to 20 BLM

• Show the students the Subtraction Change Unknown Game.
• Demonstrate how to play the game if necessary.
• Have students use the number line to help them solve the problems.

Whole Class or Groups: Summarize-

• Compare the representations they created today with the ones created for the addition problems yesterday.
• Ask students what is the same and what is different about them.

Core Math to Emphasize: Unknown change subtraction problems can be shown on a number line by counting on, counting back or subtraction.

Math Norms

49

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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Let’s Spin!

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6 + ? = 9

Spinner # Number Card

9

?

Let’s Spin!

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9

9 - ? = 6

Number Card Spinner #

?

Let’s Spin!

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- ? =

Number Card Spinner #

?

What is the same? What is different?

Lesson 5: LS 2, Day 1

Whole Class or Groups: Launch- 3 - Read Protocol

• First Read: What is the story about?
• Second Read: What are the quantities and the units in the story?
• Third Read: What math question could we ask about the story?

Independent work: Explore - Seesaw Lesson 5 - Pennies Start Unknown Pennies Student .S. .C. Pennies Three Read BLM .S. .C.

• Solve the Pennies Problem
• Use the tools and/or drawings to show your thinking.

Whole Class or Groups: Summarize-.

• Notice and Wonder
• What’s the Same? What’s Different? Compared to the Entry Task
• Core Math to Emphasize: The start can be unknown in an add to situation. Start unknown situations can be solved by counting on, counting back or subtraction.

Math Norms

56

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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1st Read

2nd Read

3rd Read

Notice and Wonder

What’s the same? What’s different?

? + 8 = 13

8 + ? = 13

Trisha has some pennies in her pocket. Joaquin gave her 8 more pennies. Now she has 13 pennies.

Trisha has 8 pennies in her pocket. Joaquin gave her some more pennies. Now she has 13 pennies.

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Lesson 6: LS 2, Day 2

Whole Class or Groups: Launch-

• Remind students of the problem they solved yesterday where the start of the problem was unknown,
• Demonstrate how to play the game using the number line
• Model counting up, counting back, and subtraction

Independent work: Explore - Seesaw Lesson 6 - Start Unknown Game, Number cards 6–20, Dice, Addition Start Unknown Game Template BLM .S. .C., Number Lines 0 to 20 BLM

• Roll the dice and use the number card
• Solve for the unknown in the equation using the number line
• Record your answer using the microphone

Whole Class or Groups: Summarize-.

• Use samples provided, or Choose 3–4 students to share a completed problem with their equation, strategy, and number line representation.
• Ask the students what is the same and what is different about the strategies and representations
• Core Math to Emphasize: Unknown start addition problems can be shown on a number line by counting on, counting back or subtraction.

Math Norms

64

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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7

DICE

7

DICE

3

7

Counting up: ? = 4

+1 +1 +1 +1

7

DICE

Counting back: ? = 4

- 1 - 1 - 1 - 1

7

DICE

X

X

X

Subtraction: 7 - 3 = 4

6 20

+4 +10

Lesson 7: LS 2, Day 3

Whole Class or Groups: Launch- 3 - Read Protocol

• First Read: What is the story about?
• Second Read: What are the quantities and the units in the story?
• Third Read: What math question could we ask about the story?
• Ask students to think about how they might set up a tape diagram or equation

Independent work: Explore - Seesaw Lesson 7 - More Pennies, Start Unknown Pennies Three Read BLM .S. .C., Start Unknown Pennies Student .S. .C., Double ten frame, Number Lines 0 to 20 BLM

• Solve using tools and drawings

Whole Class or Groups: Summarize-.Have 2–3 students share their representations and equations. Connect each part of the story problem to each part of the representation. Be sure the equation that represents the story problems is shared (? - 3 = 6). Allow students to also show the equations for solving the problem (6 + 3 = ?, 3 + 6 = ?) and distinguish between equations that represent the situation and equations we use to solve the problem.Core Math to Emphasize: The start can be unknown in a take from situation. Start unknown situations can only be solved by counting on or addition.

Math Norms

73

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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1st Read

2nd Read

3rd Read

equation:

Trisha had some pennies in her pocket. She gave 3 pennies to Joaquin. Now she has 6 pennies.

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More Pennies

Equation:

? - 3 = 6

Trisha had some pennies in her pocket. She gave 3 pennies to Joaquin. Now she has 6 pennies.

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3

6

Trisha had some pennies in her pocket. She gave 3 pennies to Joaquin. Now she has 6 pennies.

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+3

Equation: ? - 3 = 6

Trisha had some pennies in her pocket. She gave 3 pennies to Joaquin. Now she has 6 pennies.

Trisha has some pennies in her pocket. Joaquin gave her 8 more pennies. Now she has 13 pennies.

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Equation: ? + 8 = 13

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3

6

?

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?

8

13

Lesson 8: LS 2, Day 4

Whole Class or Groups: Launch-

• Show the students the Subtraction Start Unknown Game Template. Demonstrate how to play the game as necessary.
• Have students use the number line or counters with or without the double ten frame.

Independent work: Explore - Seesaw Lesson 8 - Start Unknown Game #2, Number cards 0–14, Dice, Subtraction Start Unknown Game Template BLM .S. .C., Double ten frame, Number Lines 0 to 20 BLM

• Roll the dice and use the number card
• Solve for the unknown in the equation using the number line
• Record your answer using the microphone

Whole Class or Groups: Summarize-.Notice and Wonder

• Re-enact a problem, for example, taking away 7, leaving 4. Students may recognize that they need to give the student with the question mark all of the cubes to solve the problem.

Core Math to Emphasize: Unknown start subtraction problems can be solved on a number line by counting on or addition.

Math Norms

83

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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DICE

7

DICE

7

? = 10

7

DICE

? = 10

7

DICE

• 3

? = 10

Notice and Wonder...

Lesson 9: Expert Task

Whole Class or Groups: Launch-

Show 2 – 1 = ___. Ask students what they know about this equation.

Show 2 – 1 = ____ – ____. Have students think to determine what they could put into the blanks to make the equation true.

• What does the equal sign tell us in the number sentence?
• What do the blanks tell us?
• How are the numbers on the left and right related?

Independent work: Explore - Seesaw Lesson 9 - Expert Task Subtraction Equations BLM , Number Lines 0 to 20 BLM, Double ten frame

• Make each number sentences true by deciding on the missing numbers.
• Find different combinations that will make the number sentence true.
• Complete as many pages as you can!

Whole Class or Groups: Summarize-.

Discuss visuals of different ways students might think of 8 – 3 = __ – __.

Notice and Wonder...Math Balance in Didax - Explore the equivalent relationship between the numbers on either side of an equal sign on this website with students.

• Core Math to Emphasize: There is an equivalent relationship between the numbers on either side of an equal sign.

Math Norms

91

Errors are gifts that promote discussion.

Answers are important, but they are not the math.

Talk about each other’s thinking.

Ask questions until ideas make sense.

Use multiple strategies and multiple representations.

​

SAN FRANCISCO UNIFIED SCHOOL DISTRICT

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2-1 =

2 – 1 =___ – ___

2 – 1 =__ – __

2 – 1 =__ – __

2 – 1 =__ – __

8 – 3 = __ – __.

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8 – 3 = __ – __.

8 – 3 = __ – __.

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Notice and Wonder...

Spring District Assessment Milestone Task Overview

SFUSD uses the District Assessments to support the continuous improvement work of the district. Reading Inventory (RI) is the District Assessment for reading, and two specific Math Milestone Tasks are the District Assessments for math. Aggregate student results are summarized to inform district planning and programs. Teachers use individual results to inform instruction. An individualized Student Report is uploaded to ParentVUE for families.

Ideally, assessments will encourage students to reflect on what they have learned, rather than to focus on scores, and the District Assessment is no exception. In keeping with the Graduate Profile, students are part of a community of learners who work toward graduating as critical, collaborative thinkers using multiple resources. This assessment is intended to be administered and scored as you would any other Milestone Task, after unit instruction is complete, and not treated as a high stakes assessment. The assessment should not alienate students, shift the focus away from learning towards a grade, or lead to racialized outcomes. You may administer the task whole class, in small groups, or a combination of both; in-person, online, synchronously or asynchronously.

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District Assessment Milestone Task Guidance for Teachers

Milestone Task:Trisha’s Pennies

Whole Class or Groups:Launch: 3 Read Protocol

• What’s the story about?
• What are the units and quantities?
• Act it out!

Independent work: Seesaw Milestone (Spanish), (Chinese), Trisha’s Pennies BLM .S. .C.

• Part 1: Make a tape diagram of the situation showing the unknown.
• Write an equation with an unknown to match the situation. Solve the problem.
• Part 2: Find the unknown in the equation
• Show how you know.

Students solve problems with unknowns in a variety of positions within 20.

This task is an SFUSD Math District Assessment. Chinese Slides

Whole Class or Groups: Trisha’s Pennies Rubric, Trisha’s Pennies Answer Guide Teacher, Student Work Samples and Commentary Teacher

Summarize:

Core Math to Emphasize:

• Analyzing language, making tape diagrams, and writing equations can help us make sense of word problems with unknowns.

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What is this story about?

1st Read

Trisha had 20 pennies in her pocket. She gave some to Joaquin. Then she had 7 left. How many pennies did she give Joaquin?

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What are the quantities in the situation?

2nd Read

Trisha had 20 pennies in her pocket. She gave some to Joaquin. Then she had 7 left. How many pennies did she give Joaquin?

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Act it out!

3rd Read

Trisha had 20 pennies in her pocket. She gave some to Joaquin. Then she had 7 left. How many pennies did she give Joaquin?

Trisha had 20 pennies in her pocket. She gave some to Joaquin. Then she had 7 left. How many pennies did she give Joaquin?

?

Find the unknown in the equation:

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?

11 = 8 + ?

Spring District Assessment Milestone Task Guidance

Spring District Assessment Milestone Task Guidance