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Content Area: Math
Index: 4.3C Grade 6 CPI 2
Standard: 4.3 - Patterns and Algebra
Strand: C - Modeling
Cumulative Progress Indicator: 2 - The student will draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events (e.g. changes over time; relations between quantities; & rates of change · Changes over time · Relations between quantities · Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)
Grade: 6
Sample Activities:
· Each group of students is given a Mr. or Mrs. Grasshead (i.e., a sock filled with dirt and grass seed which sits in a dish of water). They create a name for their grasshead and begin a diary, recording the number of days that have passed and the height of the grass. At the end of specified time periods, they discuss the changes in the height, the average rate of change over the time period, and the overall behavior of the grass growth. Each group makes a graph of height versus the number of days. The students note whether the graph is close to a straight line.
· Students find the number of tiles around the border of a floor 10 tiles long and 10 tiles wide by looking at smaller square floors, making a table, and identifying a pattern. They describe their pattern in words and, with assistance from the teacher, develop the expression (4 × n) + 4 for the number of border tiles needed for an n × n floor.
· Students use play money to act out the following situation and solve the problem. A man wishes to purchase a pair of slippers marked $5. He gives the shoe salesman a $20 bill. The salesman does not have change for the bill so he goes to the pharmacist next door and gets a $10 and two $5 bills. He gives the customer his change and the man leaves. The pharmacist enters shortly after and complains the $20 was counterfeit. The shoe salesman gives her $20 and gives the counterfeit bill to the FBI. How much did the shoe salesman lose?
· Students place 8 two-color chips in a paper cup and toss them ten times, recording the number of red and yellow sides showing on each toss. For each red chip that shows, they lose $1. For each yellow, they win $1. For each toss, the students write a number sentence that shows their win or loss for that toss. For example, after tossing 3 yellows and 5 reds, their sentence would read 3 - 5 = -2. Afterwards, the students look for patterns in the number sentences that they have written. They discuss these patterns and then write about them in their notebooks.
· Students read Anno's Mysterious Multiplying Jar by Mitsumasa Anno and try to analyze and represent the numerical patterns shown using variables.
· Students are asked to draw a sketch of the graph which would describe a person's distance off the ground during a ride on a ferris wheel which had a radius of 60 feet. Some students just draw a curve that looks similar to a sine curve. Others put more detail into their drawing showing the step function behavior which occurs as people get on and get off and that there are limited revolutions permitted.
· Presented with a graph showing the population of frogs in a local marsh over the past ten years, students generate hypotheses for why the curve has the shape it does. They check their hypotheses by talking with a local biologist who has studied the marsh over this time period.
· Students study which is the better way to cool down a soda, adding lots of ice at the beginning or adding one cube at a time at one minute intervals. Each student first makes a prediction and the class summarizes the predictions. Then the class collects the data, using probes and graphing calculators or computers and displays the results in table and graph form on the overhead. The students compare the graphs and write their conclusions in their math notebooks. They discuss the reasons for their results in science class.
· Students make a graph that shows the price of mailing a letter from 1850 through 1995. Some of the students begin by simply plotting points and connecting them but soon realize that the price of a stamp is constant for a period of time and then abruptly jumps up. They decide that parts of this graph are like horizontal lines. The teacher tells them that mathematicians call this a "step function"; another name for this kind of graph is a piecewise linear graph because the graph consists of linear pieces.
· Students review Mark's trip home from school on his bike. Mark spent the first few minutes after school getting his books and talking with friends, and left the school grounds about five minutes after school was over. He raced with Ted to Ted's house and stopped for ten minutes to talk about their math project. Then he went straight home. The students draw a graph showing the distance covered by Mark with respect to time. Then, with the teacher's help, the class constructs a graph showing the speed at which Mark traveled with respect to time. The students then write their own stories and generate graphs of distance vs. time and graphs of speed vs. time.
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