Teacher Lesson Guides - Equivalent Expressions

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Overview: What is a CER?

Claim Evidence Reasoning (CER) is similar to how they would defend a thought or opinion in literacy/science.

• Claim is something that you believe is true
  • What do you expect the outcome to be
  • lists, supported by evidence (prior knowledge)
• Mathematical evidence
  • Support for the claim
• Reasoning
  • Justify your response in concise sentences
  • Summarize final thinking in complete thoughts using correct mathematical vocabulary

• Show slide 2 and explain what a CER is. Ask the students if any of them have heard of CER before. Ask for one or two examples from a different context (a different subject area or in a non-academic situation)..
• Provide the non-academic example on slide 3.
  • Show the picture for about a minute and ask students to make a CER based on this photo. Do not give them a specific prompt or question, just ask them to make a claim based on what they see. Allow their creativity to guide their direction.
  • Check for understanding by having students share with a partner (Turn and Talk) for a minute each as you listen in. If time allows, have one or two groups share out.
  • Starting with a low-stakes example of a new routine helps students to understand the routine without the fear of being correct mathematically. This photo allows an entry point to the task by offering a low floor and a high ceiling.

• On slide 4, give students 1 to 2 minutes to think individually about their claims.
  • This is an opportunity for students to share/expand their thinking by engaging in a short “Turn and Talk” with a neighbor.
• On slide 5, allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)
  • After think and process time, solicit responses, adding them to the slide, the whiteboard, or chart paper as they are offered by students.
  • Work through each one of the claims and have the class as a whole sort them into similar groups. Make sure students are the ones creating the groups.
  • Have the students either agree or disagree on the claims in the groups.
  • Ask students with similar thinking to pair/group up and come up with a defense for their claim that their peers' claims are correct or incorrect..
  • Pairs/Small Group Processing Time
    • NOTE: Suggest no group larger than four. If beginning in pairs, allow for pairs to become a square to consolidate thoughts/wonderings.
    • Possible question for students: As you look at the list of claims generated by your classmates, what noticings and wonderings do you have about any one or all of them?
    • After time for pairs, then squares dialogue and noticings/wonderings, designate a person from each square to be the spokesperson. (e.g., the person wearing the most blue, having the more visible buttons, longest hair, shortest hair, etc.)
  • Whole Group Discourse
    • Have each group share out their defense. Record each defense.
    • As noticings/wonderings are offered, make notes either on the screen beside the claims, or on a board next to projected claims.
    • Return to group dialogue for 3 minutes (or less) to identify those claims that are correct, and those that need to be altered so they are also correct.
    • Do not share who is correct in their thinking, leave all of the claims up to reference back to later in the class.
  • After everything has been offered ASK:
    • What questions might you have as you look at the final thoughts and corrections offered by the class?
    • What thoughts might you have about any additional changes that should be made? Once offered, ask: How are you able to justify this change?
    • What new learning do you have now that will support your understanding of expressions?

• On slide 6, allow 1 to 2 minutes for students to generate these expressions. Start with less time and depending upon how long students are actively writing, extend as needed. Possible use of a timer.
• On slide 7, use the same process as outlined for slide 5. Allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)

• Pick different responses that stand out and have the student explain their thinking, how does what they made represent the original. (Look for keywords in their explanations such as “equivalent” and the “same quantity.”) It may be best to sequence the expressions from those which have been simplified the least to those closest to 5x+1 (but do not refer to this as “the best answer” or anything else implying that this answer is the most desirable).

**It's important to use non confrontational language- as you look at all the expressions listed, ask the students what wonders they have about what they see. You may want to record these or have students record them on paper. Avoid telling students if they are correct or incorrect and instead help facilitate conversation.**

• Students are encouraged to use their own language and understanding that allow misconceptions and misunderstandings to surface.
• The instructor should rephrase anything that needs clarification and allow the student to clarify. Other students can be asked to reword (hopefully with more accurate mathematical vocabulary) or the teacher may reword as necessary, all while honoring the thinking of students.

(Slide 8) Have students work individually or in small groups. If certain students need extra support, pull small groups for additional clarification. Make sure to check-in with groups every few minutes to see progress and help address misconceptions. Using a timer to help the instructor to keep track of time is recommended.

• Show slide 2 and remind them of the non-mathematical examples from the previous session. Do a quick review of expectations around making claims and how evidence and reasoning are communicated.
• On slide 3, give students 1 to 2 minutes to think individually about their claims.
  • This is an opportunity for students to share/expand their thinking by engaging in a short “Turn and Talk” with a neighbor.
  • The goal is to expand the multiplication problem into an addition problem.They could also lean into prior strategies for multiplying such as arrays or circles and dots. Do not push their thinking by inserting your own thinking. Ask thoughtful questions that might help them remember this strategy from elementary school.
    • “What does it mean to multiply two numbers?”
    • “How can you model your understanding?”
• On slide 4, allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)
  • After think and process time, solicit responses, adding them to the slide, the whiteboard, or chart paper as they are offered by students.
  • Work through each one of the claims and have the class as a whole sort them into similar groups. Make sure students are the ones creating the groups.
  • Have the students either agree or disagree on the claims in the groups.
  • Ask students with similar thinking to pair/group up and come up with a defense for their claim that their peers' claims are correct or incorrect..
  • Pairs/Small Group Processing Time
    • NOTE: Suggest no group larger than four. If beginning in pairs, allow for pairs to become a square to consolidate thoughts/wonderings.
    • Possible question for students: As you look at the list of claims generated by your classmates, what noticings and wonderings do you have about any one or all of them?
    • After time for pairs, then squares dialogue and noticings/wonderings, designate a person from each square to be the spokesperson. (e.g., the person wearing the most blue, having the more visible buttons, longest hair, shortest hair, etc.)
  • Whole Group Discourse
    • Have each group share out their defense. Record each defense.
    • As noticings/wonderings are offered, make notes either on the screen beside the claims, or on a board next to projected claims.
    • Return to group dialogue for 3 minutes (or less) to identify those claims that are correct, and those that need to be altered so they are also correct.
    • Do not share who is correct in their thinking, leave all of the claims up to reference back to later in the class.
  • After everything has been offered ASK:
    • What questions might you have as you look at the final thoughts and corrections offered by the class?
    • What thoughts might you have about any additional changes that should be made? Once offered, ask: How are you able to justify this change?
    • What new learning do you have now that will support your understanding of expressions?

• On slide 5, allow 1 to 2 minutes for students to generate these expressions. Start with less time and depending upon how long students are actively writing, extend as needed.

• On slide 6, use the same process as outlined for slide 4. Allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)
  • You are looking again for the same strategies used in the Do Now. This is a great way to start explaining the distributive property. Stay away from the FOIL method and try to focus on the connection to repeated addition.
  • The hope is that you get a response that shows the grouping 2x+4 added 3 times. See if there are thoughts around simplifying the expression by combining like terms. Encourage correct language in their explanations by rephrasing as you repeat each student’s response for the class. If you compare the original problem to the addition expression, ask if they notice any similarities between the two, noticings and wonderings. Try to get them to draw a conclusion that the simplified expression of 6x+12 can be attained through multiplication.
• Pick different responses that stand out and have the student explain their thinking, how does what they made represent the original. (Look for keywords in their explanations such as “equivalent” and the “same quantity.”)

**It's important to use non confrontational language- as you look at all the expressions listed, ask the students what wonders they have about what they see. You may want to record these or have students record them on paper. Avoid telling students if they are correct or incorrect and instead help facilitate conversation.**

• Students are encouraged to use their own language and understanding that allow misconceptions and misunderstandings to surface.
• The instructor should rephrase anything that needs clarification and allow the student to clarify. Other students can be asked to reword (hopefully with more accurate mathematical vocabulary) or the teacher may reword as necessary, all while honoring the thinking of students.

This is an opportunity to have students work individually or in small groups. This is also a great opportunity to pull small groups that may need more clarification. You want to have check-ins every couple of minutes to see progress and any misconceptions, the use of a timer is great for this. These could also be used as an exit ticket for understanding.

• Show slide 2 and remind them of the non-mathematical examples from the previous session. Do a quick review of expectations around making claims and how evidence and reasoning are communicated.

• On slide 3, give students 1 to 2 minutes to think individually about their claims.
  • This is an opportunity for students to share/expand their thinking by engaging in a short “Turn and Talk” with a neighbor.
• On slide 4, allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)
  • After think and process time, solicit responses, adding them to the slide, the whiteboard, or chart paper as they are offered by students.
  • Work through each one of the claims and have the class as a whole sort them into similar groups. Make sure students are the ones creating the groups.
  • Have the students either agree or disagree on the claims in the groups.
  • Ask students with similar thinking to pair/group up and come up with a defense for their claim that their peers' claims are correct or incorrect..
  • Pairs/Small Group Processing Time
    • NOTE: Suggest no group larger than four. If beginning in pairs, allow for pairs to become a square to consolidate thoughts/wonderings.
    • Possible question for students: As you look at the list of claims generated by your classmates, what noticings and wonderings do you have about any one or all of them?
    • After time for pairs, then squares dialogue and noticings/wonderings, designate a person from each square to be the spokesperson. (e.g., the person wearing the most blue, having the more visible buttons, longest hair, shortest hair, etc.)
  • Whole Group Discourse
    • Have each group share out their defense. Record each defense.
    • As noticings/wonderings are offered, make notes either on the screen beside the claims, or on a board next to projected claims.
    • Return to group dialogue for 3 minutes (or less) to identify those claims that are correct, and those that need to be altered so they are also correct.
    • Do not share who is correct in their thinking, leave all of the claims up to reference back to later in the class.
  • After everything has been offered ASK:
    • What questions might you have as you look at the final thoughts and corrections offered by the class?
    • What thoughts might you have about any additional changes that should be made? Once offered, ask: How are you able to justify this change?
    • What new learning do you have now that will support your understanding of expressions?
  • Look for 2m+2m+2m+12+12+12 or 3m+3m+18+18. Any variations of this start moving the thinking towards factoring.

• On slide 5, allow 1 to 2 minutes for students to generate these expressions. Start with less time and depending upon how long students are actively writing, extend as needed.

• On slide 6, use the same process as outlined for slide 4. Allow students to make one mathematical claim at a time and record them on a whiteboard, in the slide provided, or on an interactive tool like Padlet. As you collect responses, make no comments, nor give non-verbal indicators as to the correctness of the claims. (If students are not yet comfortable with sharing their ideas with a class, have them all capture their claims anonymously on sticky notes or note cards.)
  • Look for students who make connections to factoring through repeated addition. The first task is to draw connections between equivalency of different representations and to show factoring.
  • The hope is that you get a response that shows that 3 groups of (a+b) is equal to 3a+3b. See if there are thoughts around simplifying the expression by combining like terms. Encourage correct language in their explanations by rephrasing as you repeat each student’s response for the class. Try to get them to draw a conclusion that the two strip diagrams are equivalent, and that one can be created from the other if the pieces are rearranged.
• Pick different responses that stand out and have the student explain their thinking, how does what they made represent the original. (Look for keywords in their explanations such as “equivalent” and the “same quantity.”)
• Task 2 on slide 7 is a real word application for factoring. Allow 1 to 2 minutes for students to generate these expressions. Start with less time and depending upon how long students are actively writing, extend as needed. Possible use for a timer.
• On slide 8, use the same process as outlined for slide 4.
  • You are looking for similar strategies from the Do Now. Look for common factors through expanding the expression.

**It's important to use non confrontational language- as you look at all the expressions listed, ask the students what wonders they have about what they see. You may want to record these or have students record them on paper. Avoid telling students if they are correct or incorrect and instead help facilitate conversation.**

• Students are encouraged to use their own language and understanding that allow misconception and misunderstanding to surface.
• The instructor should rephrase anything that needs clarification and allow the student to clarify. Other students can be asked to reword (hopefully with more accurate mathematical vocabulary) or the teacher may reword as necessary, all while honoring the thinking of students.

This is an opportunity to have students work individually or in small groups. This is also a great opportunity to pull small groups that may need more clarification. You want to have check-ins every couple of minutes to see progress and any misconceptions, the use of a timer is great for this. These could also be used as an exit ticket for understanding.