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Wagons West!

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Summary

Students will describe the challenges the pioneers faced as they began their journey west. Students will understand and apply the elements of the historical fiction genre. Students will use historical fiction to learn more about a time period.

Coin Type(s)

  • Quarter

Coin Program(s)

  • 50 State Quarters

Objectives

  • Students will describe the challenges the pioneers faced as they began their journey west.
  • Students will understand and apply the elements of the historical fiction genre.
  • Students will use historical fiction to learn more about a time period.

Major Subject Area Connections

  • Language Arts
  • Social Studies

Grades

  • Fourth grade
  • Fifth grade
  • Sixth grade

Class Time

Sessions: Five
Session Length: 45-60 minutes
Total Length: 151-500 minutes

Groupings

  • Whole group
  • Small groups
  • Individual work

Background Knowledge

Students should have a basic knowledge of:

  • Pioneers
  • Westward expansion
  • Journal writing
  • Timeline
  • Genre
  • Historical fiction

Terms and Concepts

  • Setting
  • Character
  • Plot
  • Point of View

Materials

  • Nebraska quarters (optional)
  • 1 overhead projector
  • 1 class map of the United States
  • Chart paper
  • Markers
  • 1 overhead transparency of each of the following:
    • “Nebraska Map”
    • “Background Information” page
    • “Nebraska Quarter Reverse” page (or photocopy)
    • “Can You Analyze This?” worksheet
    • “Monumental Measurements” page
  • Locate multiple copies of a historical fiction text about pioneers, such as:
    • Roughing It on the Oregon Trail by Diane Stanley
    • Don’t Know Much About the Pioneers by Kenneth C. Davis
    • Growing Up in Pioneer America: 1800 To 1890 (Our America) by Judith Pinkerton Josephson
    • Children of the Westward Trail by Rebecca Stefoff
    • You Wouldn’t Want to Be an American Pioneer! by Jacqueline Morley
    • If You Traveled West in a Covered Wagon by Ellen Levine
    • Words West: Voices of Young Pioneers by Ginger Wadsworth
    • A Pioneer Sampler: The Daily Life of a Pioneer Family in 1840 by Barbara Greenwood
    • Daily Life in a Covered Wagon by Paul Erickson
    • Pioneer girl: Growing up on the Prairie by Andrea Warren
  • Highlighters
  • Copies of the following worksheets:
    • “Preview the Text”
    • “Can You Analyze This?”
    • “Reading Guide”
    • “Journal Writing Guide”
    • “Westward Journals” (optional)
  • Copies of texts about pioneers to add to a class library, such as:
    • Daily Life in a Covered Wagon by Paul Erickson
    • Frontier Schools and School Teachers by Ryan P. Randolph
    • If You Traveled West in A Covered Wagon by Ellen Levine
    • My Face to the Wind: The Diary of Sarah Jane Price, A Prairie Teacher by Jim Murphy
    • A Pioneer Sampler: The Daily Life of a Pioneer Family in 1840 by Barbara Greenwood
    • Prairie Songs by Pam Conrad
    • Roughing It on the Oregon Trail by Dane Stanley
    • Stories of Young Pioneers in Their Own Words by Violet T. Kimball
    • Winter Thunder by Marie Sandoz
    • Words West: Voices of Young Pioneers by Ginger Wadsworth
    • You Wouldn’t Want To Be an American President by Jacqueline Morley

Preparations

  • Gather Nebraska quarters (1 per student) (optional)
  • Make copies of the following worksheets:
    • “Preview the Text” (1 per student)
    • “Can You Analyze This?” (1 per student)
    • “Reading Guide” (1 set of 3 pages per student)
    • “Journal Writing Guide” (1 per student)
    • “Westward Journals” (optional) (1 per student)
  • Make an overhead transparency of each of the following:
    • “Nebraska Quarter Reverse” page
    • “Nebraska Map”
    • “Background information” worksheet
    • “Can You Analyze This?” worksheet
    • “Monumental Measurements” page
  • Locate multiple copies of a historical fiction text about pioneers (see examples under “Materials”).
  • Gather copies of a text about pioneers for a class library (see examples under “Materials”).
  • Note that students may need an additional session to finish reading the text.

Worksheets and Files

Lesson plan, worksheet(s), and rubric (if any) at www.usmint.gov/kids/teachers/lessonPlans/pdf/319.pdf.

Session 1

  1. Describe the 50 State Quarters® Program for background information, if necessary, using the example of your own state, if available. Locate Nebraska on a classroom map. Note its position in relation to your school’s location.
  2. Display the “Nebraska Quarter Reverse” overhead transparency or photocopy. Option- ally, distribute actual quarters. Have the students identify the images in this coin design, including Chimney Rock, the pioneers, and the wagon. Lead a class discussion regarding the images.
  3. Display the “Nebraska Map” overhead transparency and point out the Platte River Valley, the Oregon and California Trails, and the city of Omaha on the map. Remind the students that Chimney Rock is located west of Courthouse and Jailhouse Rock. Locate it and mark it on the “Nebraska Map.” Collect the quarters, if used.
  4. Tell the students that many drawings of Chimney Rock were made by artists, and most pioneers mentioned it in their diaries. Travelers reported that it was visible 40 miles away. Discuss this distance with the students. Talk about a store or other recognizable landmark that is 40 miles away from your school. Tell the students that Chimney Rock is said to have “the appearance of a haystack with a pole running far above its top” (by General Joel Palmer, leading a surveying party in 1845).
  5. Display the “Monumental Measurements” overhead transparency and look briefly at the relative sizes of the different monuments shown. Tell the students that Chimney Rock was one of the most recognizable landmarks along the Oregon Trail. For the pioneers, it marked the end of the prairies and the approach to the Rocky Mountains. Talk to the students about the pioneers traveling along the Oregon Trail. Compare the size of Chimney Rock (325 feet from tip to base and 120 feet for the spire) with other landmarks like the Eiffel Tower (985 feet) and Statue of Liberty (305 feet).
  6. Display the “Background Information” overhead transparency. Discuss the information with the students. Using the “Nebraska Map,” point out the main routes the pioneers traveled and the importance and difficulty of traveling during that time in history.
  7. On a piece of chart paper, start a concept web about the pioneer travels west. Add the student’s thoughts and other ideas covered by the class discussion to the web.
  8. Explain to the students that they will be reading historical fiction in small groups to learn more about the westward journey of pioneers in the mid-1800s. Define “historical fiction” as a story based on historical fact. All reading will be done during class time.
  9. Divide the class into small groups and choose (or let students choose) a text to read.
  10. Distribute a “Preview the Text” worksheet to the students.
  11. Have the students get into their small groups to preview the text and complete the worksheet.
  12. As a class, add any new content to the web.
  13. Collect the completed “Preview the Text” worksheet.

Session 2

  1. Review the concept web and the information discussed from Session 1 with the students.
  2. Display the “Can You Analyze This?” overhead transparency and distribute a “Can You Analyze This?” worksheet to each student. Review the content of the worksheet. Encourage the students to highlight key information during the discussion.
  3. Model analyzing a story line with the students using a simple story that is familiar to the students such as “The Three Little Pigs.” Use the “Can You Analyze This?” overhead transparency as a guide for analyzing the story. Tell the students they will use this worksheet as a guide later. Encourage the students to take notes on the discussion.
  4. Distribute the “Reading Guide” pages to the students. As a class, review the directions. Discuss and complete the vocabulary definitions at the top of the worksheet (see answer key). The students will use the definitions to help them complete the rest of the worksheet. Review the rest of the questions with the class.
  5. Have the students get into their small groups and begin reading the chosen texts. Circulate among the groups and check for comprehension of the texts.
  6. When there is about 10 minutes left in the class period, bring the class back together. As a class, discuss overall themes and content from the texts.
  7. Review the web from the previous session and add any new information to the concept web.
  8. Collect the “Reading Guide” worksheets.

Session 3

  1. Have the students get into their small groups from the previous session. Redistribute the “Reading Guide” worksheets to the students.
  2. Have the students continue reading from the chosen text. The students should continue to fill in information on their “Reading Guide” worksheets. Circulate among the groups.
  3. Once the students have completed the reading, discuss some of the student responses including the characters, setting, and plot on the “Reading Guide” worksheets.
  4. Review and add any new information to the concept map.
  5. Collect the “Reading Guide” worksheets.

Session 4

  1. Display the overhead transparency of the “Nebraska Quarter Reverse” page.
  2. As a class, discuss the details of the image on the coin. Point out the wagon and style of dress. Discuss the sequence of challenges that pioneers may have faced on their journey westward regarding animals, food, and shelter. On a piece of chart paper, create a list of challenges and record the students’ responses.
  3. Display the challenges in the classroom for student reference. Provide time for the stu- dents to add any information from the concept web and chart to their “Reading Guide” worksheets.
  4. Collect the “Reading Guide” worksheets.
  5. Distribute the “Writing Guide” and “Journal Writing Guide” worksheets to the students.
  6. Review the criteria of the journals ensuring that the students understand the following:
    • The journal entries that they will write are to be from the perspective of a member of a pioneer family traveling the trail.
    • There needs to be a sketch of Chimney Rock included in one of the entries.
    • Each entry must address at least one of the challenges often faced along the trail.
  7. Allow enough time for the students to create their journal entries, using the worksheets and charts from previous sessions as a resource.
  8. Have the students share their journal entries with others.
  9. Collect all of the students’ worksheets.

Differentiated Learning Options

  • Provide books on tape.
  • Allow students to dictate their journal entries to a scribe.

Enrichments/Extensions

Once the journal entries are completed, have the students create a map showing the trail their characters took or a symbolic timeline to highlight the journey. Share and display the maps and timelines in the classroom.

  • Take anecdotal notes about the students’ participation in class discussions.
  • Evaluate students’ journal entries for integration of the historical information taught.
There are no related resources for this lesson plan.

Discipline: Language Arts
Domain: W.4 Writing
Grade(s): Grade 4
Cluster: Text Types and Purposes
Standards:

  • W.4.1. Write opinion pieces on topics or texts, supporting a point of view with reasons and information.
    • Introduce a topic or text clearly, state an opinion, and create an organizational structure in which related ideas are grouped to support the writer’s purpose.
    • Provide reasons that are supported by facts and details.
    • Link opinion and reasons using words and phrases (e.g., for instance, in order to, in addition).
    • Provide a concluding statement or section related to the opinion presented.
  • W.4.2. Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
    • Introduce a topic clearly and group related information in paragraphs and sections; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.
    • Develop the topic with facts, definitions, concrete details, quotations, or other information and examples related to the topic.
    • Link ideas within categories of information using words and phrases (e.g., another, for example, also, because).
    • Use precise language and domain-specific vocabulary to inform about or explain the topic.
    • Provide a concluding statement or section related to the information or explanation presented.
  • W.4.3. Write narratives to develop real or imagined experiences or events using effective technique, descriptive details, and clear event sequences.
    • Orient the reader by establishing a situation and introducing a narrator and/or characters; organize an event sequence that unfolds naturally.
    • Use dialogue and description to develop experiences and events or show the responses of characters to situations.
    • Use a variety of transitional words and phrases to manage the sequence of events.
    • Use concrete words and phrases and sensory details to convey experiences and events precisely.
    • Provide a conclusion that follows from the narrated experiences or events.

Discipline: Language Arts
Domain: W.4 Writing
Grade(s): Grade 4
Cluster: Production and Distribution of Writing
Standards:

  • W.4.4. Produce clear and coherent writing in which the development and organization are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3 above.)
  • W.4.5. With guidance and support from peers and adults, develop and strengthen writing as needed by planning, revising, and editing. (Editing for conventions should demonstrate command of Language standards 1-3 up to and including grade 4.)
  • W.4.6. With some guidance and support from adults, use technology, including the Internet, to produce and publish writing as well as to interact and collaborate with others; demonstrate sufficient command of keyboarding skills to type a minimum of one page in a single sitting. 

Discipline: Language Arts
Domain: W.5 Writing
Grade(s): Grade 4
Cluster: Production and Distribution of Writing
Standards:

  • W.5.4. Produce clear and coherent writing in which the development and organization are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3.)
  • W.5.5. With guidance and support from peers and adults, develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. (Editing for conventions should demonstrate command of Language standards 1-3 up to and including grade 5 here.)
  • W.5.6. With some guidance and support from adults, use technology, including the Internet, to produce and publish writing as well as to interact and collaborate with others; demonstrate sufficient command of keyboarding skills to type a minimum of two pages in a single sitting.

Discipline: Language Arts
Domain: W.5 Writing
Grade(s): Grade 4
Cluster: Text Types and Purposes
Standards:

  • W.5.1. Write opinion pieces on topics or texts, supporting a point of view with reasons and information.
    • Introduce a topic or text clearly, state an opinion, and create an organizational structure in which ideas are logically grouped to support the writer’s purpose.
    • Provide logically ordered reasons that are supported by facts and details.
    • Link opinion and reasons using words, phrases, and clauses (e.g., consequently, specifically).
    • Provide a concluding statement or section related to the opinion presented.
  • W.5.2. Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
    • Introduce a topic clearly, provide a general observation and focus, and group related information logically; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.
    • Develop the topic with facts, definitions, concrete details, quotations, or other information and examples related to the topic.
    • Link ideas within and across categories of information using words, phrases, and clauses (e.g., in contrast, especially).
    • Use precise language and domain-specific vocabulary to inform about or explain the topic.
    • Provide a concluding statement or section related to the information or explanation presented.
  • W.5.3. Write narratives to develop real or imagined experiences or events using effective technique, descriptive details, and clear event sequences.
    • Orient the reader by establishing a situation and introducing a narrator and/or characters; organize an event sequence that unfolds naturally.
    • Use narrative techniques, such as dialogue, description, and pacing, to develop experiences and events or show the responses of characters to situations.
    • Use a variety of transitional words, phrases, and clauses to manage the sequence of events.
    • Use concrete words and phrases and sensory details to convey experiences and events precisely.
    • Provide a conclusion that follows from the narrated experiences or events.

Discipline: Language Arts
Domain: W.6 Writing
Grade(s): Grade 4
Cluster: Production and Distribution of Writing
Standards:

  • W.6.4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3 above.)
  • W.6.5. With some guidance and support from peers and adults, develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. (Editing for conventions should demonstrate command of Language standards 1–3 up to and including grade 6.)
  • W.6.6. Use technology, including the Internet, to produce and publish writing as well as to interact and collaborate with others; demonstrate sufficient command of keyboarding skills to type a minimum of three pages in a single sitting.

Discipline: Language Arts
Domain: L.4 Language
Grade(s): Grade 4
Cluster: Conventions of Standard English
Standards:

  • L.4.1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.
    • Use relative pronouns (who, whose, whom, which, that) and relative adverbs (where, when, why).
    • Form and use the progressive (e.g., I was walking; I am walking; I will be walking) verb tenses.
    • Use modal auxiliaries (e.g., can, may, must) to convey various conditions.
    • Order adjectives within sentences according to conventional patterns (e.g., a small red bag rather than a red small bag).
    • Form and use prepositional phrases.
    • Produce complete sentences, recognizing and correcting inappropriate fragments and run-ons.
    • Correctly use frequently confused words (e.g., to, too, two; there, their).
  • L.4.2. Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.
    • Use correct capitalization.
    • Use commas and quotation marks to mark direct speech and quotations from a text.
    • Use a comma before a coordinating conjunction in a compound sentence.
    • Spell grade-appropriate words correctly, consulting references as needed.

Discipline: Language Arts
Domain: L.5 Language
Grade(s): Grade 4
Cluster: Conventions of Standard English
Standards:

  • L.5.1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.
    • Explain the function of conjunctions, prepositions, and interjections in general and their function in particular sentences.
    • Form and use the perfect (e.g., I had walked; I have walked; I will have walked) verb tenses.
    • Use verb tense to convey various times, sequences, states, and conditions.
    • Recognize and correct inappropriate shifts in verb tense.
    • Use correlative conjunctions (e.g., either/or, neither/nor).
  • L.5.2. Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.
    • Use punctuation to separate items in a series.
    • Use a comma to separate an introductory element from the rest of the sentence.
    • Use a comma to set off the words yes and no (e.g., Yes, thank you), to set off a tag question from the rest of the sentence (e.g., It’s true, isn’t it?), and to indicate direct address (e.g., Is that you, Steve?).
    • Use underlining, quotation marks, or italics to indicate titles of works.
    • Spell grade-appropriate words correctly, consulting references as needed.

Discipline: Language Arts
Domain: L.6 Language
Grade(s): Grade 4
Cluster: Conventions of Standard English
Standards:

  • L.6.1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.
    • Ensure that pronouns are in the proper case (subjective, objective, possessive).
    • Use intensive pronouns (e.g., myself, ourselves).
    • Recognize and correct inappropriate shifts in pronoun number and person.
    • Recognize and correct vague pronouns (i.e., ones with unclear or ambiguous antecedents).
    • Recognize variations from standard English in their own and others' writing and speaking, and identify and use strategies to improve expression in conventional language.
  • L.6.2. Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.
    • Use punctuation (commas, parentheses, dashes) to set off nonrestrictive/parenthetical elements.
    • Spell correctly.

Discipline: Language Arts
Domain: RL.4 Reading: Literature
Grade(s): Grade 4
Cluster: Craft and Structure
Standards:

  • RL.4.4. Determine the meaning of words and phrases as they are used in a text, including those that allude to significant characters found in mythology (e.g., Herculean).
  • RL.4.5. Explain major differences between poems, drama, and prose, and refer to the structural elements of poems (e.g., verse, rhythm, meter) and drama (e.g., casts of characters, settings, descriptions, dialogue, stage directions) when writing or speaking about a text.
  • RL.4.6. Compare and contrast the point of view from which different stories are narrated, including the difference between first- and third-person narrations.

Discipline: Language Arts
Domain: RL.4 Reading: Literature
Grade(s): Grade 4
Cluster: Integration of Knowledge and Ideas
Standards:

  • RL.4.7. Make connections between the text of a story or drama and a visual or oral presentation of the text, identifying where each version reflects specific descriptions and directions in the text.
  • RL.4.8. not applicable to literature.
  • RL.4.9. Compare and contrast the treatment of similar themes and topics (e.g., opposition of good and evil) and patterns of events (e.g., the quest) in stories, myths, and traditional literature from different cultures.

Discipline: Language Arts
Domain: RL.4 Reading: Literature
Grade(s): Grade 4
Cluster: Key Ideas and Details
Standards:

  • RL.4.1. Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text.
  • RL.4.2. Determine a theme of a story, drama, or poem from details in the text; summarize the text.
  • RL.4.3. Describe in depth a character, setting, or event in a story or drama, drawing on specific details in the text (e.g., a character’s thoughts, words, or actions).

Discipline: Language Arts
Domain: RL.5 Reading: Literature
Grade(s): Grade 4
Cluster: Craft and Structure
Standards:

  • RL.5.4. Determine the meaning of words and phrases as they are used in a text, including figurative language such as metaphors and similes.
  • RL.5.5. Explain how a series of chapters, scenes, or stanzas fits together to provide the overall structure of a particular story, drama, or poem.
  • RL.5.6. Describe how a narrator’s or speaker’s point of view influences how events are described.

Discipline: Language Arts
Domain: RL.5 Reading: Literature
Grade(s): Grade 4
Cluster: Integration of Knowledge and Ideas
Standards:

  • RL.5.7. Analyze how visual and multimedia elements contribute to the meaning, tone, or beauty of a text (e.g., graphic novel, multimedia presentation of fiction, folktale, myth, poem).
  • RL.5.8. not applicable to literature.
  • RL.5.9. Compare and contrast stories in the same genre (e.g., mysteries and adventure stories) on their approaches to similar themes and topics.

Discipline: Language Arts
Domain: RL.6 Reading: Literature
Grade(s): Grade 4
Cluster: Craft and Structure
Standards:

  • RL.6.4. Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of a specific word choice on meaning and tone.
  • RL.6.5. Analyze how a particular sentence, chapter, scene, or stanza fits into the overall structure of a text and contributes to the development of the theme, setting, or plot.
  • RL.6.6. Explain how an author develops the point of view of the narrator or speaker in a text.

Discipline: Language Arts
Domain: RL.6 Reading: Literature
Grade(s): Grade 4
Cluster: Integration of Knowledge and Ideas
Standards:

  • RL.6.7. Compare and contrast the experience of reading a story, drama, or poem to listening to or viewing an audio, video, or live version of the text, including contrasting what they “see” and “hear” when reading the text to what they perceive when they listen or watch.
  • RL.6.8. not applicable to literature.
  • RL.6.9. Compare and contrast texts in different forms or genres (e.g., stories and poems; historical novels and fantasy stories) in terms of their approaches to similar themes and topics.

Discipline: Language Arts
Domain: W.4 Writing
Grade(s): Grade 4
Cluster: Research to Build and Present Knowledge
Standards:

  • W.4.7. Conduct short research projects that build knowledge through investigation of different aspects of a topic.
  • W.4.8. Recall relevant information from experiences or gather relevant information from print and digital sources; take notes and categorize information, and provide a list of sources.
  • W.4.9. Draw evidence from literary or informational texts to support analysis, reflection, and research.
    • Apply grade 4 Reading standards to literature (e.g., “Describe in depth a character, setting, or event in a story or drama, drawing on specific details in the text [e.g., a character’s thoughts, words, or actions].”).
    • Apply grade 4 Reading standards to informational texts (e.g., “Explain how an author uses reasons and evidence to support particular points in a text”).

Discipline: Language Arts
Domain: W.5 Writing
Grade(s): Grade 4
Cluster: Research to Build and Present Knowledge
Standards:

  • W.5.7. Conduct short research projects that use several sources to build knowledge through investigation of different aspects of a topic.
  • W.5.8. Recall relevant information from experiences or gather relevant information from print and digital sources; summarize or paraphrase information in notes and finished work, and provide a list of sources.
  • W.5.9. Draw evidence from literary or informational texts to support analysis, reflection, and research.
    • Apply grade 5 Reading standards to literature (e.g., “Compare and contrast two or more characters, settings, or events in a story or a drama, drawing on specific details in the text [e.g., how characters interact]”).
    • Apply grade 5 Reading standards to informational texts (e.g., “Explain how an author uses reasons and evidence to support particular points in a text, identifying which reasons and evidence support which point[s]”).

Discipline: Language Arts
Domain: W.6 Writing
Grade(s): Grade 4
Cluster: Research to Build and Present Knowledge
Standards:

  • W.6.7. Conduct short research projects to answer a question, drawing on several sources and refocusing the inquiry when appropriate.
  • W.6.8. Gather relevant information from multiple print and digital sources; assess the credibility of each source; and quote or paraphrase the data and conclusions of others while avoiding plagiarism and providing basic bibliographic information for sources.
  • W.6.9. Draw evidence from literary or informational texts to support analysis, reflection, and research.
    • Apply grade 6 Reading standards to literature (e.g., “Compare and contrast texts in different forms or genres [e.g., stories and poems; historical novels and fantasy stories] in terms of their approaches to similar themes and topics”).
    • Apply grade 6 Reading standards to literary nonfiction (e.g., “Trace and evaluate the argument and specific claims in a text, distinguishing claims that are supported by reasons and evidence from claims that are not”).

Discipline: Mathematics
Domain: All Communication
Cluster: Instructional programs from kindergarten through grade 12 should enable all students to
Grade(s): Grades K–12
Standards:

  • organize and consolidate their mathematical thinking through communication 
  • communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
  • analyze and evaluate the mathematical thinking and strategies of others; and
  • use the language of mathematics to express mathematical ideas precisely.