Unit 1: Graphing and Application Teacher: R Knittle Subject Area: Algebra 1 Grade: 9 Year: 2009-2010 Content Learning Outcomes Big Ideas Assessments Primary Target Standards Students will be able to: STATE STANDARDS Ordered Pairs Scatter Plots Correlation Discrete / Continuous Relationships Data from tables and equations Apply technology to data information Graph coordinate information Interpret graphs Create stories from graphs Interpret intercepts and slope in general terms Build equations from tables Formulate questions that can be addressed with data. Use mathematical models to represent and understand quantitative relationships Recognize and manipulate rates of change Recognize patterns through tables Formative: Homework practice Classroom talk Graphing calculator partner work Summative: Graph interpretation assignment Creating a scatter plot Interpreting a correlation Building an equation from a table Knot Tying Lab (Data trend to graph and equation with explanations) A1.N.4 Use estimation to judge the reasonableness of results of solutions to problems. A1.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions. A1.P.4 Translate between different representations of functions and relations: graphs, equations, and tabular. A1.P.5 Demonstrate an understanding of the relationship between various representations of a line. Determine a line s slope, intercepts from a graph or equation. A1.D.2 Approximate a line of best fit. Use technology when appropriate.
Lesson: Overview Building Ordered Pairs: From Data Points to Scatter Plots Subject/Grade Level: 9 Lesson Time: 60 Lesson Synopsis Students will begin to interpret information and organize it in a mathematical way. Students will translate between random data and build toward a correlation (if possible). Big Idea Data representing, organizing, and understanding State Standards A1.N.4 Use estimation to judge the reasonableness of results of solutions of problems. A1.P.3 Demonstrate an understanding of relations and functions. A1.P.4 Translate between different representations of functions. Determine a line s slope, intercepts from a graph or equation. A1.D.2 Approximate a line of best fit. Use technology when appropriate. Learning Outcomes - Students will be able to: Describe ways of collecting and organizing data. Recognize the characteristics of a scatter plot and correlation. Interpret graphs and correlations. Key Cognitive Strategies Problem Solving Research Interpretation Reasoning R.Knittle Building Ordered Pairs: From Data Points to Scatter Plots
Materials Needed Chart paper, overhead projector, overhead markers, transparencies, and a good set of data points.. Another component will be converting word relationships into scatter plots.. Copies of graphs that show correlation and scatter plots (at least three graphs). Seven different correlations that students will use to create scatter plots (along with printed titles of each correlation to post on walls). Lesson Activities Step 1 Literacy Group Questioning Collaborative Group Work Step 2 Assign groups and pass out correlation and scatter plot information. Put students in mixed ability groups of 3. Have the students read aloud in groups (one at a 10 min Assign each student a number in the group. time) and underline important information as they are being read to. The Ones will read to the group through the first graph, the Twos will read to the group through the second graph and the Threes will finish the document. Remind students to underline important data as the graph is being read. As you circulate among the groups, notice what is being underlined. Create a wall chart of features that are important for students to remember. Help students determine characteristics of correlation graphs. Have students generate a list of important data when they finish reading aloud. Have the entire group of students share their findings and generate a word wall. R.Knittle Building Ordered Pairs: From Data Points to Scatter Plots
Collaborative Group Work Instruct students to read three correlation graphs and determine three or more important characteristics of them in general. (e.g. positive directions right and up, intervals evenly spaced, labels, independent versus dependent axis, title, origin, etc.) Create another wall chart that lists characteristics of correlation graphs. Have students look at the first three correlation graphs to determine important characteristics and have the students share their findings with the class. Have the students answer: What do you see that makes this correlation graph special, so that you can create your own graph correctly the next time? 10 min Step 3 Writing to Learn Write summaries of correlation graphs. Have students create notes on correlation Have students generate notes from the word walls 10 Min from the word walls. Important questions that they find important about the characteristics of must be clarified by the students correlation graphs. Have students work themselves. Ask the students What notes individually for three minutes, then have them can you create from this information on compare notes within their group. correlation? Extend the discussion to Ask some students to read their responses to the correlation coefficient between -1 and 1, class, and ask others to summarize what they have and what would the words no correlation, written. strong positive correlation and strong negative correlation look like? Step 4 Writing to Learn Questioning Give each group the seven different correlations from which they will attempt to create a scatter plot (translating words to graphs). Assign four correlations per group, and Have students generate a graph from one of the 20 min rotate the correlations that the students will four assigned correlations and create a rough produce. Hang the titles of each correlation scatter plot. After three minutes, ask students to R.Knittle Building Ordered Pairs: From Data Points to Scatter Plots
Collaborative Group Work on the wall, so that students can place their graphs under the correct title. tape their scatter plots under the appropriate title. When all graphs are displayed, ask the students to summarize what they notice about each correlation in their notes. Lesson Closing Writing to Learn Closing Teacher Activity Closing Student Activity Time Assign an exit ticket (a small piece of paper Give students five minutes to write a summary of 10 Min for a brief assignment), to the class. what creates a strong positive correlation, a strong negative correlation, and no correlation. Along with Collect exit tickets as students leave class. it, ask students to create a graph to match each summary (with a minimum of fifteen data points). To further this, ask students what a weak positive or negative correlation would look like. Allow two minutes for class discussion of the different possibilities before starting the writing assignment. Assessments of Learning Outcomes Class observation and discussion Poster paper presentation Small group discussion Exit ticket R.Knittle Building Ordered Pairs: From Data Points to Scatter Plots
Lesson: Overview Creating and Finding Scatter Plots from Real World Data Subject/Grade Level: 9 Lesson Time: 60 min Lesson Synopsis Students will create scatter plots from real world data of right-handed and left-handed students in the class. This Right-Hand/Left- Hand Graph assignment was adapted from http://math.rice.edu/~lanius/algebra/rightleft.html Big Idea Create a graph and interpret real world data. State Standards A1.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions. A1.P.4 Translate between different representations of functions and relations: graphs, equations, and tabular. A1.D.2 Approximate a line of best fit. Use technology when appropriate. Learning Outcomes - Students will be able to: Create a scatter plot from information. Generate details about the scatter plot. Research data to generate more information. Key Cognitive Strategies Problem Solving Research Interpretation Reasoning Creating and Graphing Scatter Plots From Real-World Data
Materials Needed Teachers will need an overhead projector, transparencies, over-head markers, and grid paper for graphing left-hand and right-hand data (one grid for each student). See an example of a grid at http://math.rice.edu/~lanius/algebra/rightleft.html Students will need graph paper and colored pencils. Notebooks are essential. Lesson Activities Step 1 Pass out grid paper to students and give them thirty seconds to fill in as many boxes in the first column as they can with their first initial using their right hand. Give students a twenty second break, then Have students fill in the first column of the grid 5 min give them thirty seconds to fill in as many with their first initial using their right hand, and boxes as they can with their left hand in the after a twenty second break, give them thirty second column again writing their first seconds to fill in the second column of the grid initial. with their first initial using their left hand. This may be done as a group activity, but works best without students comparing what they are doing. At the bottom of the paper, have students total the number of right hand initials they formed and the total of left hand initials they formed. Step 2 Gather from the students the tally of the right hand initials each student formed and the tally of the left hand initials each student formed. Create a data table on the board or Have students report their tallies and copy the 5 min overhead projector with three columns: complete data table from the board in their student name, right hand tally, and left notebooks. hand tally. Fill in the information from each student. Creating and Graphing Scatter Plots From Real-World Data
. Step 3 Questioning Ask the students what they notice about the data. Record the answers on the board or Have the students discuss what they notice about 5 min overhead projector (e.g. more students the data. have higher right hand numbers, some students have equal amounts, some are much higher numbers ). Step 4 Collaborative Group Work Pair students with a partner next to them and have each pair create a scatter plot of the final right hand and left hand tallies. Give students graph paper to design a Have the students create scatter plots of the final 15 Min scatter plot. Encourage students to recall right hand and left hand tallies. the necessary characteristics of a scatter plot and recall how to graph a single ordered pair. To create consistenty across pairs, ask each pair to record the right hand data on the horizontal axis. As well, have the students use axes with the same intervals. Once students have recorded all the data, have them create a special line on their graphs by creating points in a different color where all the right hand data matches the left hand data. Creating and Graphing Scatter Plots From Real-World Data
Step 5 Collaborative Group Work Writing to Learn Questioning Ask the students what they notice about the scatter plots and have students answer questions in their notebooks. Ask the students the question, What do Have students answer questions in their notebooks. 10 Min you notice? and record answers on the board. Other questions to ask (especially if students are finishing at different times, these can be posted on the board): Connect the points that are a different color. What is significant of these points? What is significant about the points below the line? What is significant about the points above this line? Step 4 Collaborative Group Work Writing to Learn Give the students five minutes to generate other questions to investigate. Have students work with their partners to Ask students What other types of information 10 min record any questions they think they can could you gather using this same idea? research. Have them share their lists of Creating and Graphing Scatter Plots From Real-World Data
questions and post on the board. Have students work with their partners to generate further research questions. Lesson Closing Questioning Closing Teacher Activity Closing Student Activity Time Help students generate a positive research Students choose a question that they and their 10 min question to do for homework. One or two partner will investigate for homework. different questions may be addressed, i.e.: Are you right footed or left footed? Age and height? Body distance correlations? Make it clear that students must address these questions within their research: How many participants will they find for the survey? How will they graph their data together? What further information do they need? Assessments of Learning Outcomes Partner scatter plot Low stakes writing assignment Research question and information generated Creating and Graphing Scatter Plots From Real-World Data
Lesson: Building Slope Concepts Overview Subject/Grade Level: 9 Lesson Time: 60 min Lesson Synopsis Students will see rate of change as slope through tables, graphs, and equations. Big Idea Determining slope from tables, graphs, and equations State Standards A1.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of the function. A1.P.5 Determine a line s slope and x-and y- intercepts from its graph or linear equation. Learning Outcomes - Students will be able to: Determine slope from tabular information and from given equations. Recognize rates of change as a means of slope. See the connections between tables, graphs, equations, slope, and intercepts. Key Cognitive Strategies Problem Solving Interpretation Reasoning Materials Needed Poster paper, markers, graph paper, overhead, transparency, and worksheets Family Car Trip Scenario Activity (see step 1) Sample tables and graphs for activities to recognize slope and intercepts; one table should have missing information for the students Building Slope Concepts
to find and fill in (see step 2) Practice sheet for students to work from equations, generate tables as a group of solutions, and then determine the slopes and intercepts (see step 3) Lesson Activities Step 1 Collaborative Group Work Questioning Students work with a partner to design a graph of a family car trip. Distribute activity to the students and have 15 min them work with a partner to produce the graph. Have an overhead of the product, so as students finish, partners go to the overhead to produce a component of the scenario. Have students talk through the picture of the graph, relating a word scenario, and as a peer edits, have them make sure the graph matches the correlation information from earlier in the unit. Help students to see the difference between the two scenarios. Step 2 Questioning Students work with tables and graphs to recognize slope and intercepts. Continue with partner work by having 15 min students see the connection between tables and graphs. Review the concepts from correlation to build further connections. Provide multiple methods of finding slope through constant first difference, making triangles between data points, or change in Have students look for slope and intercepts, making connections between tables, graphs, and slope. Have students share methods of finding slope. Have students fill in missing data from a table, either by finding answer on graph, or through constant differences. Building Slope Concepts
y over change in x, which is an important way for students to see tables as a means of finding any slope. Step 3 Questioning Writing to Learn Lesson Closing Writing to Learn Working from equations, students generate tables as a group of solutions, and then determine the slopes and intercepts. Pass out copies of practice sheet to Have students fill in the table to find various 25 Min students. Have an overhead of the practice solutions to the given equations. As they create the sheet. entire table, have students determine the slope. Have students find the slope by the student using one of the three means previously discussed, and have students find data points as a part of the effort. Looking for fractional answers creates difficult learning. Discuss with students the ideas necessary to find reasonable data points. Have students answer the question at the end of this sheet: What do you notice? Introduce students to the phrase the zeroeth term. Have students find the y-intercept from this term. Have students find other ways to generate slope and y-intercept. Have students answer the question What do you notice?, by seeing slope and y-intercept within the equation when it is solved for Y. Ask students to graph the equations so that they see these five vocabulary words together: equation, table, graph, slope and y-intercept. Closing Teacher Activity Closing Student Activity Time Give a writing prompt for the students to Have the students answer the writing prompt in 5 min write in their notes. Convey that there is a their notes: Describe the connections seen today value in writing notes to the students. This between equations, tables, graphs, slope, and writing activity will show which concepts intercepts. What have you seen that makes sense to are clear to students and which are not. you, and what have you seen that didn t make Building Slope Concepts
After giving the prompt, discuss it with entire class. sense? Assessments of Learning Outcomes Low stakes writing notes Worksheets that are built into student notebook Building Slope Concepts
Lesson: Overview Creating Equations to Further Model Relationships Subject/Grade Level: 9 Lesson Time: 60 min Lesson Synopsis Students will synthesize building equations from expressions, tables, and graphs. Big Idea Modeling math connections State Standards A1.N.4 Use estimation to judge the reasonableness of results and solutions to problems. A1.P.1 Describe, complete, extend, and analyze linear patterns. A1.P.5 Demonstrate an understanding of linear relationships. Determine a line s slope and intercepts from its graph or equation. Explain the significance of positive, negative, zero or undefined slope. Learning Outcomes - Students will be able to: Create tables, graphs and equations of math models. Find the slope and intercepts of the information. Interpret information and extend information. Key Cognitive Strategies Problem Solving Interpretation Reasoning Precision Materials Needed Chart paper, markers, rulers, graph paper, calculators (optional) Modeling Graphs and Equations
Skip-Jump-Words-Protocol (see step 1) Three-problem worksheet (see step 2) Information on the relationship between a cricket s chirps and temperature (see step 3) Lesson Activities Step 1 Questioning Review the Skip-Jump-Words-Protocol with students. (See document attached). Students will begin to see relationships with numbers. Tables are to be extended with patterns. Use the Skip-Jump-Words-Protocol to Have students work to answer the phrase For 15 Min review word phrases to expressions and every human year it is alive, a dog ages 7 years. explore connections between words, tables, Have students build the table and walk through graphs, and expressions. Use this table to to developing the expression. discuss first differences, and the connection, to slope. You may provide another word phrase for students if necessary. Step 2 Collaborative Group Work Questioning Group students in heterogeneous groups of three and distribute a three-problem worksheet. Allow ten minutes to work on all three Have students work individually within groups for 35 min problems simultaneously. ten minutes to solve one of three problems. (10 min) Modeling Graphs and Equations
Writing to Learn After ten minutes, assign each group one problem to present on chart paper. Enforce the concept that these skills will be on the assessment in the next class. During the presentation, have students take notes in class. Circulate among the groups at all times to maintain focus, but do not answer the groups questions. Let the groups ask and answer each other s questions. If a presentation ends and a group in the audience has a different method, give them extra class points. Ask students to present one of the three problems to the entire class on poster paper. Have students work together on how best to present and explain multiple methods of solving the problem. Give more credit for more methods. Have students volunteer any different methods they used at the end of presentations. individu al work (15 min) poster preparati on (10 min) poster presentat ion Step 3 Questioning Have students practice finding the temperature given the number of a cricket s chirps in a minute. Post information on crickets. Direct Have students use two data points to create an 5 Min students to find the dependent and equation, then generate future predictions. Should independent variable, generate an equation, time allow, have students discuss all new graph, find intercepts (provided students vocabulary words used in the unit and explain their can explain what they mean), and predict meaning. If time does not allow, assign this as the temperature if they hear 100 chirps in a homework. minute. Lesson Closing Closing Teacher Activity Closing Student Activity Time Writing to Learn Post an exit ticket asking the students to Have students answer the exit ticket question: 5 min Modeling Graphs and Equations
voice concerns they have over the class problems of the day. Collect the answers, so students know which types of problems to expect on quiz the next day. Which of the three group problems did you understand the best and why? Which of the three group problems did you least understand and why? What do you need help with before the next assessment? Assessments of Learning Outcomes Collaborative Group Work Practice problem Group Quiz in Two Days (30 minutes for three problems) Exit Tickets Modeling Graphs and Equations
Lesson: Knot Tying Lab (A) Overview Subject/Grade Level: 9 Lesson Time: 60 min Lesson Synopsis Students will learn to connect graphing calculators with real world information to graph and interpret equations of lines. Big Idea Creating connections between graphs, data, and tables State Standards A1.N.4 Use estimation to judge the reasonableness of solutions. A1.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions. A1.P.4 Translate between different representations of functions and relations: graphs, equations, and tabular. A1.P.5 Determine a line s slope and x- and y- intercepts from its graph or from a linear equation. A1.D.1 Select, create and interpret graphical representations for a set of data and use appropriate ways to communicate information about the data. A1.D.2 Approximate a line of best fit (trend line) given a set of data. Use technology when appropriate. Learning Outcomes - Students will be able to: See data generating equations of lines. Learn connections between slope and parts of an equation. Learn connections between intercepts and parts of an equation. Make connections between tables, graphs, and equations. Key Cognitive Strategies Problem Solving Knot Tying Lab (A)
Interpretation Reasoning Precision Materials Needed Ropes of varying thickness and length, meter sticks with centimeter measure, ruler, tape measure, graph paper, tables, and graphing calculators. Lab information and lab papers Lesson Activities Step 1 Collaborative Group Work Group students into groups of two and distribute ropes and all other materials. NOTE: Design the pairs before hand, Have the Engineers pick up all material for the lab 5 min matching students with mixed abilities, but -- a rope, ruler, meter stick, graph paper, and take into account personalities as well. You calculator. want people to work with others they don t know, but be careful to not create a pair Have the Data/Question Recorders pick up the lab that cannot work together. papers and begin to pre-read the information. Describe the roles to the students, and have them choose which role they will play. One is the Data/Question Recorder, and the other is the Engineer. Engineer reads all details, generates conversation, pushes for clarity. Data/Question Recorder records all questions the pair have and all data useful during the lab. This person Give students five minutes to preview the lab and look for details that might be confusing. Have the students look for clarifying questions. Knot Tying Lab (A)
maintains precise information for the entire lab. Once the preview time has expired, make sure students understand the directions. Give students one minute to ask any clarifying questions. Once this is over, have the students answer their own questions together as a team. Step 2 Clarify roles and routines and describe the overall plan for the class. Show students a rope length and how to measure the entire length to within a tenth of a centimeter. Demonstrate tying a knot and discuss making a gentle knot. Have students record the new distance. Describe that they will continue to follow this method, but must accurately record all information. Explain that every group will have different numbers, and they need to be precise as possible. Remind students that they MUST read the lab and respond as a team to each question asked. Have students listen carefully to the overall plan of the lab, and understand how their role is vital to their team s success. 5 min Note also they do not need to complete twenty knots, but should be able to generate at a minimum more than seven. Knot Tying Lab (A)
Step 3 Collaborative Group Work Complete the lab. Circulate around the room, helping Have students generate all the information possible 40 Min students with continual clarifying questions. for the lab: build all tables, find first differences, If too many partners ask the same question, utilize the graphing calculator, and respond to pull the class back together to clarify the questions in the packet. directions for all students. Most groups will get through to the graphing calculator portion of the lab. Encourage all groups (as you circulate) to get to that point, helping where necessary. Lesson Closing Collaborative Group Work Writing to Learn Questioning Closing Teacher Activity Closing Student Activity Time Bring students back to focus, although they 10 min may be at different points in the lab. Collect all materials from the students. Keep lab with rope. Lay the foundation for the definitions of correlation, domain, range, slope, dependent and independent variable for the writing activity.. Have the students respond in their journals to the questions: Describe in words what was happening to your rope section, and why it was happening. What was happening every time you created a new knot? What type of a correlation is occurring, and how do you know? In other words, describe the correlation. Have students respond to the questions in their notebooks, and discuss in class to generate thinking for the next class. Knot Tying Lab (A)
Assessments of Learning Outcomes Completed (partially) lab Discussion will expose connections to slope, domain, range, y-intercept and x-intercept Knot Tying Lab (A)
Lesson: Knot Tying Lab (B) Overview Subject/Grade Level: 9 Lesson Time: 60 min Lesson Synopsis Students will learn to connect graphing calculators with real world information to graph and interpret equations of lines. Big Idea Creating connections between graphs, data, and tables State Standards A1.N.4 Use estimation to judge the reasonableness of solutions. A1.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions. A1.P.4 Translate between different representations of functions and relations: graphs, equations, and tabular. A1.P.5 Determine a line s slope and x- and y- intercepts from its graph or from a linear equation. A1.D.1 Select, create and interpret graphical representations for a set of data and use appropriate ways to communicate information about the data. A1.D.2 Approximate a line of best fit (trend line) given a set of data. Use technology when appropriate. Learning Outcomes - Students will be able to: See data generating equations of lines. Learn connections between slope and parts of an equation. Learn connections between intercepts and parts of an equation. Make connections between tables, graphs, and equations. Key Cognitive Strategies Problem Solving Knot Tying Lab (B)
Interpretation Reasoning Precision Materials Needed Ropes of varying thickness and length, meter sticks with centimeter measure, ruler, tape measure, graph paper, tables, and graphing calculators. Lab information and lab papers from previous lesson. Graph paper and assessment to find how the students understand the vocabulary of each learning outcome. Lesson Activities Step 1 Re-group students into groups of two. Distribute ropes and all other materials to students. Have students locate their data from the previous lesson and load in a graphing calculator. Have students load their data into their graphing calculator and continue working through the lab. 5 min All students must complete through answering the question What does the y- intercept really mean? Step 2 Collaborative Group Work Questioning Have students use graphic calculators to understand scatter plots. Circulate through the room, guiding Have students continue to work on the lab, 20 min partners to further use technology to generating equations of lines, with intercepts understand what is occurring. The found, and the true meaning of slope, domain, Knot Tying Lab (B)
graphing calculators will show students the relation of data points and also give them a variety of information. Help students to determine the reasonableness of data. Should students complete lab early, give them acollege-readiness question to deal with. Extensions are important to look for, for example, ask how many knots of a certain size can you create in a rope of a certain size? range, reasonableness, and intercept. Students will see connections between table information, graphs, and equations. Step 3 Writing to Learn Give students a low stakes assessment. Distribute graph paper and assessment. Have students complete a low-stakes assessment to 30 min This is not a detailed assessment, just a determine their understanding of the vocabulary of check in to find how the students each learning outcome. understand the vocabulary of each learning outcome. Lesson Closing Writing to Learn Closing Teacher Activity Closing Student Activity Time Propose the closing question to the class. Give students a two-minte free-write period to 5 min answer the closing question: Generate a list of responses. List all vocabulary words you ve come across Knot Tying Lab (B)
through the lab and define them. You will have two minutes to list as many as you can. Review your notes or lab sheet if necessary. Remember, you have only two minutes to complete this assignment. Have students share their responses. Assessments of Learning Outcomes Low stakes assessment on vocabulary Classroom talk on equations of lines, specifically of the form y=mx+b (slope-intercept form) Knot Tying Lab (B)