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Unit 2, Lesson 15

Off the Line

Extra Support Materials for Algebra 1

15.1

Warm-up

Standards Alignment

Building On

Addressing

Instructional Routines

Aspects of Mathematical Modeling Estimation

Materials

None

Activity Narrative

The purpose of this Warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also to help students develop a deeper understanding of the meaning of standard units of measurement. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

Launch

Display the image. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right. Invite students to record these numbers and to write a short explanation for the reasoning behind their estimates.

Activity

None

<p>Photograph, 9 nickels and 7 picture hangers on a scale, weight 52 grams.</p>

How much does a nickel weigh?

Record an estimate that is:

too low about right too high
Explain your reasoning.

Student Response

to access Student Response.

Activity Synthesis

The goal of this discussion is for students to analyze the reasonableness and accuracy of their estimates.

Ask a few students to share their estimates and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these responses for all to see in an ordered list or on a number line. Add the least and greatest estimates to the display by asking, “Is anyone’s estimate less than ? Is anyone’s estimate greater than ?” If time allows, ask students, “After this discussion, does anyone want to revise their estimate?”

In a following activity, students will work to find the answer.

Encourage students to think about and discuss the accuracy of their estimates and the estimates that are displayed by asking questions, such as:

"Is the actual value inside the range of values?"
"Is it toward the middle?"
"How variable are the estimates?"
"What are the sources of error?"

Consider developing a method to record a “snapshot” of the estimates and the actual value so that over time students can track their progress as estimators.

15.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this activity is for students to recognize that multiplying an equation by a value results in the same solution. This will be useful when transforming equations for the elimination method of solving systems of equations in the supported Algebra 1 lesson. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work out their problems independently and collaborate with one another when they do not arrive at the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner's work or by explaining their reasoning.

Launch

Arrange students in groups of two. In each group, ask students to decide who will work on column A and who will work on column B. If students are unfamiliar with the row game structure, demonstrate the protocol before they start working.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts, for example two or three rows at a time. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Social-Emotional Functioning

15.3

Activity

Instructional Routines

MLR3: Critique, Correct, Clarify

Materials

To Gather

Coins Number cubes

Activity Narrative

The mathematical purpose of this activity is for students to get a chance to follow through on a plan of how someone might reason through a contextual situation that can be modeled by a system of linear equations. As students come to understand solving equations as a process of reasoning about equivalent statements, they benefit from getting to interpret the reasoning of others.

This will benefit students when they attempt to solve systems of linear equations on their own in their Algebra 1 class.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time, and check whether your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.

Solve each equation for .

row	column A	column B
1
2
3
4
5
6

Activity Synthesis

The purpose of this discussion is to help students synthesize and generalize what Jada and Priya were doing as they explored different combinations of objects whose weights they could determine, which would help them find the true weights of the objects.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response explaining Jada and Priya’s processes by correcting errors, clarifying meaning, and adding details.

Display this first draft:
Jada and Priya found out the weight of each object by making it so the amount of one of the objects was the same on each scale and then adding the scales together.
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Here are some questions for discussion:

"When did they add or subtract what was on the scales? Why?" (When they knew the weight of the same number of items as are on the scale, they could subtract to find the weight of the remaining items.)
"When did they multiply what was on the scales? Why?" (When they knew the weight of some items and multiplying would make it match the number of items on the other scale. This would let them subtract the weights to find out the weight of the remaining items.)
"What does this have to do with solving systems of equations?" (There are two situations with two unknown weights of objects.)
"Can you think of a move that Jada or Priya could have made that would not have helped them?" (Adding the weights at the beginning to find the combined weight of all the items would not have been useful.)

If time allows, ask students to pick one of Jada's or Priya’s moves that they thought was crucial, and to explain why it was so helpful in finding the weight of one object on the scale.

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Lesson 15

Off the Line

Warm-up

Standards Alignment

Building On

Addressing

5.NF.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Building Toward

Standards Alignment

Building On

Addressing

8.EE.C.7

Building Toward

HSA-REI.C.5

Standards Alignment

Building On

Addressing

HSA-REI.A

Building Toward

HSA-REI.C.6