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Intervention Support. Developing understanding and long-term retention requires conceptual teaching Developing fluency requires practice including corrective feedback and frequent cumulative review. Proportional Reasoning. Ratios and Proportions.
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Developing understanding and long-term retention requires conceptual teaching • Developing fluency requires practice • including corrective feedback and frequent cumulative review
Ratios and Proportions • Understanding ratios and using proportional thinking is the most important set of math concepts we teach in middle school • Ratios grow out of fractions and lead into linear functions • Proportional thinking is used all the time in everyday life – unit prices, miles per gallon, converting measurement units, etc. etc. etc.
Ratios At the school dance, there were 4 boys for every 3 girls. Draw a picture of what 4 to 3 looks like to you. If there were 133 students at the dance, how many were boys and how many were girls? How might students solve this?
One important meaning of fraction • In a bag of M&Ms that contains 40 candies, 8 are red. Every bag, no matter how large, has this same ratio of red candies. What fraction of candies are red? • How many red candies would you expect to find in a bag of 25 M&Ms? • If you pulled out 5 M&Ms from any bag, should you expect to get one red candy? Why? See Recommendations for Effective Fraction Instruction for K-8
Multiplicative thinking NOT 5:1 6:2 7:3 8:4 9:5 _:8 5:1 10:2 15:3 20:4 25:5 40:8 For every 5 M&Ms, there is 1 red M&M. You see the connection to slope, right?
Constant of proportionality In a bag of M&Ms that contains 40 candies, 8 are red. Every bag, no matter how large, has this same ratio of red candies. What fraction of candies are red? • Let y = the total number of candies • Let x = the number of red candies What equation relates y and x? (This is another way of showing the multiplicative relationship.) or You see the connection to slope, right?
A large bottle of shampoo contains 40 fluid ounces. The bottle recommends that you use 1¼ fluid ounces per application. How many shampoo washes can you do with this bottle? This is a proportion problem where the constant of proportionality is 1¼:1. What various ways might you represent this? How do you read this equation, in the context of the problem?
5 Ways to Solve Proportion Problems • Find a unit rate • Use a table • Use a tape diagram • Set up and solve a proportion equation • Make a graph
Solve these problems using each of the strategies. • If 3 pounds of hamburger feeds 12 people, how many pounds are needed for 30 people? • A restaurant recipe calls for 10 cups of flour to make 4 cakes, but you want to make only 1 cake. How much flour do you need? • If we can drive 150 miles in 3 hours, how many hours will it take us to drive 400 miles? • Find a unit rate • Use a table • Use a tape diagram • Set up and solve a proportion equation • Make a graph
If 6 cats can catch 6 mice in 6 minutes, how many cats does it take to kill 100 mice in 50 minutes? • Think about an army of 6 cats. They can do their job at the rate of 6 mice in 6 minutes. What’s the unit rate? • If you want to kill 100 mice in 50 minutes, what’s that unit rate? So how big does your army of cats need to be?
Categorical Data Two friends want to start a restaurant, but they don’t know what kind. Should it be a breakfast café, a lunch and dinner grill, or a fine dining restaurant open after 4 pm. To decide, they need to do some research, so they’ve come to our class to ask us: What meal of the day is your favorite? Make a bar graph and circle graph to show the data. (Bar graphs are learned in 1st-3rd grade. Making circle graphs require knowing how to find and draw the angles.)
6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Statistical Questions • What’s the average age of middle school teachers who attend professional development workshops? • How large are extended families in Mid-Michigan, as indicated by the number of cousins for any family member? • What is the distribution of birthdates through the year?
Process of Statistical Analysis • Formulate the question • Collect the data • Represent and analyze the data • Interpret and present our results
Numerical Data 18, 18, 20, 22, 22, 22, 22, 22, 23, 25, 25, 25, 25, 25, 26, 26, 27, 27, 30, 30, 32, 32, 37 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Create your own number lines: http://themathworksheetsite.com/numline.html
http://www.shodor.org/interactivate/activities/PlopIt/ Dot Plot Using Squares 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Histogram http://www.shodor.org/interactivate/activities/Histogram/ Cluster the data into bands 17-20 3 21-23 6 24-26 7 27-29 2 30-32 4 33-35 0 36-38 1 18, 18, 20, 22, 22, 22, 22, 22, 23, 25, 25, 25, 25, 25, 26, 26, 27, 27, 30, 30, 32, 32, 37
6.SP.5 Summarize numerical data sets in relation to their context, such as: a. Reporting the number of observations. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered. Box Plot An outlier is often defined as any data that is more or less than 1.5 IQR above 3rd quartile or below 1st quartile. In this case, IQR = 5, 1.5*5=7.5, 27+7.5 = 34.5 http://www.shodor.org/interactivate/activities/BoxPlot/
Working with Our Data Summarize and describe the distribution, following 6.SP.4 and 6.SP.5. Decide which display is most appropriate for your data set. Find the most appropriate measure of center and give a quantitative measure of variability. Describe any overall patterns, etc.
Comparing Samples How do these two samples compare?
7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
18 7.26 18 7.26 20 5.26 22 3.26 22 3.26 22 3.26 22 3.26 22 3.26 23 2.26 25 0.26 25 0.26 25 0.26 25 0.26 25 0.26 26 0.74 26 0.74 27 1.74 27 1.74 30 4.74 30 4.74 32 6.74 32 6.74 37 11.74 79.3 3.45 abs difference between value and mean Mean = 25.26 sum of deviations from the mean mean of deviations (MAD)
Comparing Samples Difference of medians: 6 MAD of Adults: 3.45 Difference of medians is almost twice the variability. 7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
http://www.illustrativemathematics.org/ College Athletes PDF is on our wiki
Bivariate Data 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
Animal body weight vs. brain size From Illustrative Mathematics
Bivariate Categorical Data 8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
7.SP.6-7 • Roll a number cube 60 times and count the number of times it comes up either 1 or 6. Compare your data with others at the table. If you rolled a number cube 1000 times, how many times would you expect it to come up either 1 or 6? • How often would it come up exactly that number of times? • Plot the number of times it came up either 1 or 6 for each group in the room on a line plot.
7.SP.8 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
If the probability that it will rain on Saturday is 50% and the probability that it will rain on Sunday is 50%, what’s the probability that it will rain sometime during the weekend? Can you use an organized list, table, or tree diagram? What’s a simulation that can answer this question, using a coin flip?
