7.RP.A.2. Recognize and represent proportional relationships between quantities. (M) a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total Connections: In the first section (7.RP.A.1) students use various methods for finding unit rates and examining equivalent fractions. In this section, they make the connection between proportional relationships and graphs, tables, and function rules (in the form y=kx). They should be able to state what a point (x,y) or (1,r) means in the context of a proportional relationship on a graph. The connection is made to measurement when students convert between different units of measure and write that relationship in the form of a table or graph. Notes: Students’ understanding of the multiplicative reasoning used with proportions continues from 6th grade. Students determine if two quantities are in a proportional relationship from a table. Ratio (rate) tables are used to build equivalent ratios/rates. Proportional relationships can be represented symbolically (equation), graphically (coordinate plane), in a table, in diagrams, and verbal descriptions. Proportional relationships that are written in the form of an equation should be written in the form y=kx. Proportions that are written in the form of two equivalent fractions should be reserved for unit 3. Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships. Special attention should be spent on analyzing the points (0, 0) and (1, r) where r is the unit rate. Ratio and proportional reasoning strategies can be extended and applied to multi-step ratio and percent problems. Background Knowledge: Teaching Student-Centered Mathematics, Van de Wallle Pages 154-178 Lesson AIMS Proportional Reasoning: Proportional Practice (a) Rectangle Ratios (a) Recognizing Proportions (a) Patterns in Equivalent Fractions (a) Mixing Measures (d) The Shadow Knows (c) AIMS Looking at The Thick of Things Shadow Lines Reflection Takes a Turn
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats. Students should be able to identify which fractions will terminate (the denominator of the fraction in reduced form only has factors of 2 and/or 5) This happens because only factors with denominators containing factors of 2 and/or 5 can be rewritten in tenths, hundredths, thousandths, etc… Examples: 1. Examine the family of equations. What patterns do you see? Create a model and context for each of the products. Write and model the family of equations related to 3 × 4 = 12. 2. The distributive property states that (−1)x+1x is the same as (−1+1)x, and this is 0. It follows that (−1)x is the same as −x. Explain why, then use similar reasoning to explain why (−x)y is the same as −(xy). By the way, is it correct to say, −x is a negative number ? 3. A recipe calls for cups of flour. You only have a cup measuring cup. How many times will you need to fill the measuring cup? Source: North Carolina Public Schools (d) (d)
of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (S) Use proportional reasoning to make estimates relating to a population on the basis of a sample. Inferences a population are only valid if the sample is random and representative. Having multiple samples for the same population allows for gauging the variation of estimates or predictions. Examples: Below is the data collected from two random samples of 100 students regarding student’s school lunch preferences. Make at least two inferences based on the results. Source: Arizona Academic Content Standards Illustrative Mathematics Task: 7.SP.B.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 Notes: This is the students’ experience with comparing two data sets. Students build on their understanding of graphs, mean, median, Mean Absolute Deviation (M.A.D.) and interquartile range from 6th grade. Mean absolute deviation is an element of a data set that is the absolute difference between that element and a given point. Measures of center can be used to compare data and measure variability between data sets. Use the mean (or other measures of center) to find the difference between the centers as a measure of variability. Data displays are used to visually compare data sets and draw informal comparative inferences. Box plots are way to show measures of variability such as the range (Other data displays may highlight other measures of variability). Use observations to about mean differences between two or more samples to make conjectures about the populations from which the samples were taken. Background Information: Lesson Mean Absolute Deviation (Glencoe Lesson Seed) Illustrative Mathematics Task:
7.SP.C.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (S) 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. c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Notes: Students use tree diagrams, frequency tables, and organized and simulations to determine the probability of compound events. Probabilities are useful for predicting what will happen over the long run. Using theoretical probability, students predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the experimental data give realistic estimates of the probability of an event but are affected by sample size. Examples: Students conduct a bag pull experiment. A bag contains 5 marbles. There is one red marble, two blue marbles and two purple marbles. Students will draw one marble without replacement and then draw another. What is the sample space for this situation? Explain how you determined the sample space and how you will use it to find the probability of drawing one blue marble followed by another blue marble. Suppose, over many years of records, a river generates a spring flood about 40% of the time. Based on these records, what is the chance that it will flood for at least three years in a row sometime during the next five years? Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn at random, what is the probability that you will draw the letters F-R-E-D in that order? What is the probability that your “word” will have an F as the first letter? Source: Arizona Academic Content Standards Background Knowledge: Lesson Ideas: Memory Games Illustrative Mathematics Tasks: (a, b)
7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (A) Notes: Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be expressed as Pi. Building on these understandings, students generate the formulas for circumference and area. Develop and utilize formulas for the area and circumference of circles. Examples: Lesson Ideas: AIMS Looking at Lines: Functions in Circles