Probability is one of the major shifts in the Common Core State Standards for Mathematics. In previous years, probability concepts have started in early elementary and developed in complexity through the years. The foundational probability concepts now reside in Grade 7. This calls for another filter adjustment in the classroom.
So...let's get this probability party started! What's your favorite probability concept to teach? Why? What is a looming concern you have about probability? And what obstacles have you experienced when assessing student understanding of probability concepts in the past (CCSS or not)? Solutions?
Join the discussion in the comment section below!
12.04.2013
11.23.2013
Solving Percent Problems
Since percents are ratios, the concept of percents is deeply rooted in proportional reasoning. The Common Core State Standards require students to use proportional relationships to solve percent problems. Teachers can approach percent problems by using proportional relationships with ratios or equations; this post outlines a path using ratios.
Before teaching percent change, students should have prior knowledge of percents. Students should be able to find a percent of a quantity as a rate per 100. And students should also be able to find the whole when given a part and the percent. Both of these concepts are included in CCSS 6.RP.A.3; however, teachers may prefer to informally assess these skills prior to introducing percent change. The "Calculate This!" activity requires students to determine the fraction, decimal, and percent represented by the shaded portion of a figure. (Note: The document is organized to be printed double-sided with answers on the back of the corresponding figure.) The "Estimate This!" activity requires students to estimate percent problems and sort the estimates into four categories including less than 50%, between 50% and 100%, between 100% and 200%, and more than 200%.
Percent change is a common real-world application. Percent increase applications include tax, markups, fees, gratuities and commissions. Percent decrease applications include discounts or markdowns. A procedural approach to the concept of percent change can quickly overwhelm students either through the series of steps or in the confusion of how to duplicate a "shortcut" on their own. For example, a 5% discount on a $45 hoodie purchased with 9.5% sales tax could have a process emphasis with the following steps: 5% of $45=$2.25; $45-$2.25=$42.75; 9.5% of $42.75=$4.06; $42.75+$4.06=$46.81 total OR 95% of $45=$42.75; 109.5% of $42.75=$46.81 total. And when students are asked to determine the discounted price or the amount of tax, they struggle differentiating the steps to extract the specific solution. This is a great reminder that process must first be grounded conceptually. Perhaps a geometric investigation using color tiles with focus on percent change in area could provide students with a visual reference point when encountering the hoodie problem.
One common percent change misconception that becomes problematic in the midst of percent error is the misunderstanding that the percentage of increase from an original area to a new area is the same percentage of decrease from the new area back to the original area. Teachers can informally assess students via a journal prompt and then provide remediation as necessary by using the RallyCoach structure for Kagan cooperative learning.
Percent error shows how inaccurate an experimental measurement is. If the experimental measurement is more than the actual measurement, describe the inaccuracy by what percent increase the experimental (new) measurement is compared to the actual (original) measurement. If the experimental measurement is less than the actual measurement, describe the inaccuracy by what percent decrease the experimental (new) measurement is compared to the actual (original) measurement. The Mars Research Lab Activity poses a suspected issue in one plant that prompts students to analyze samples to determine percent error for each color of candy within the sample. This lab activity can be used independently or as the research for a culminating project.
These activities highlight the Common Core State Standard 7.RP.A.3 included in MATH-7.
Before teaching percent change, students should have prior knowledge of percents. Students should be able to find a percent of a quantity as a rate per 100. And students should also be able to find the whole when given a part and the percent. Both of these concepts are included in CCSS 6.RP.A.3; however, teachers may prefer to informally assess these skills prior to introducing percent change. The "Calculate This!" activity requires students to determine the fraction, decimal, and percent represented by the shaded portion of a figure. (Note: The document is organized to be printed double-sided with answers on the back of the corresponding figure.) The "Estimate This!" activity requires students to estimate percent problems and sort the estimates into four categories including less than 50%, between 50% and 100%, between 100% and 200%, and more than 200%.
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| Fraction, Decimal, and Percent Representations |
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| Percents Sorting Activity |
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| Percent Change Investigation |
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| Percent Change Misconceptions |
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| Mars Research Lab Activity |
11.22.2013
Multi-Step Ratio Problems
Proportional relationships are the major focus in the Common Core State Standards for grade 7 math. And this requires teachers to be intentional about setting the stage to make connections across the concepts included in the Ratios and Proportional Relationships domain.
The Desert Run team activity, adapted from a practice problem in the Math Course 2 textbook by McDougal Littell, centers around three animals and their top speeds. When comparing the running distances provided in the activity, students may initially think the greyhound rules the desert. However, further investigation is required to compare distance per unit of time because the snapshot distances are collected in different time intervals. This activity prompts students to determine which animal is the fastest and how far each animal has traveled after 11 seconds.
This activity prompts rich discussion with an opportunity to critique the work of others and make connections between rates and proportions. A record sheet is included for convenient use with the Fan-N-Pick or Showdown structures for Kagan cooperative learning.
The most popular proportional relationship is found in the formula d=rt. If your students are unfamiliar with the formula, lead them to it by asking them to create a table of values that model the relationship between the distance in feet that the greyhound travels over time in seconds and then write the equation that models the relationship. Extend their typical y=ax equation to better model the situation...let "y" be represented by "d" for distance in feet, let "a" be represented by "r" for rate, and let "x" be represented by "t" for time in seconds. Consider the following discussion path to help students connect equivalent rates to the equation for proportional relationships.
The concepts discussed in the Desert Run team activity build the foundation needed for multi-step ratio problems. Modeling the sample "Reading Three Books" Type 1 task from PARCC, this Solving Multi-Step Ratio Problems team activity uses the Simultaneous RoundTable structure for Kagan cooperative learning. This activity assesses conceptual understanding of unit rates and its application in a multi-step ratio problem.
These activities highlight Common Core State Standards 7.RP.A.2b, 7.RP.A.2c, and 7.RP.A.3 included in MATH-7.
The Desert Run team activity, adapted from a practice problem in the Math Course 2 textbook by McDougal Littell, centers around three animals and their top speeds. When comparing the running distances provided in the activity, students may initially think the greyhound rules the desert. However, further investigation is required to compare distance per unit of time because the snapshot distances are collected in different time intervals. This activity prompts students to determine which animal is the fastest and how far each animal has traveled after 11 seconds.
- Students may calculate the unit rate for the speed of each animal to determine which one is fastest. And then they may continue to use the unit rates to determine the distance each animal travels in 11 seconds.
- Students may set up a proportion to determine the distance after 11 seconds and use those results to declare which animal is fastest.
- Students may use the unit rate to determine the fastest animal and then set up a proportion to determine the distances after 11 seconds.
This activity prompts rich discussion with an opportunity to critique the work of others and make connections between rates and proportions. A record sheet is included for convenient use with the Fan-N-Pick or Showdown structures for Kagan cooperative learning.
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| Rates and Proportions Activity |
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| Connecting Rates and Proportions |
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| Solving Multi-Step Ratio Problems |
11.02.2013
Proportional vs. Nonproportional Relationships
Middle school students are naturally concrete thinkers. Using manipulatives to represent situations is one method to extend students from concrete thinking to abstract thinking. The following activity prompts students to use color tiles to model relationships, transfer that model to a numerical table, and then differentiate the characteristics of proportional and nonproportional relationships within the context.
This investigation provides your students with a common reference point for proportional relationships. A student record sheet is available to support the three "homework vs. video games" scenarios in the investigation. And suggestions for class discussion are included in the "Teacher Notes" section.
The big ideas that are characteristic of proportional relationships can be summarized in a Frayer Model. Post this graphic while students are completing targeted practice with the key concepts. Then ask students to create a Frayer Model at the conclusion of the unit. You will be surprised by the detail they add to their own graphics!
This investigation highlights Common Core State Standard 7.RP.A.2 that is included in MATH-7.
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| Investigation |
This investigation provides your students with a common reference point for proportional relationships. A student record sheet is available to support the three "homework vs. video games" scenarios in the investigation. And suggestions for class discussion are included in the "Teacher Notes" section.
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| Proportional Relationships Record Sheet |
The big ideas that are characteristic of proportional relationships can be summarized in a Frayer Model. Post this graphic while students are completing targeted practice with the key concepts. Then ask students to create a Frayer Model at the conclusion of the unit. You will be surprised by the detail they add to their own graphics!
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| Frayer Model |
This investigation highlights Common Core State Standard 7.RP.A.2 that is included in MATH-7.
9.16.2013
Decimal Form of a Rational Number
In the post CCSS Call for Rigor, we contemplated the necessity of implementing concept teaching that reveals the process in order to meet the standards.
Assuming students have prior knowledge of the definition of a rational number, how would you address the 7.NS.A.2d standards?
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.Since students have used the standard algorithm for division in 6th grade (6.NS.B.2), would you take a procedural approach or a conceptual approach?
Process...given fractions use long division to observe the results of decimals that terminate or repeat. Is that doable? Certainly. What's the problem? The students haven't learned why some fractions terminate and others repeat; which means the students haven't connected those results with the standard algorithm. And they would most likely be unable to discuss the results without trudging through long division every single time.
Concept...given fractions develop a geometric model.
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| Geometric Discovery Model |
Students then use the model to connect part-to-whole with the decimal expansion to determine which fractions terminate and which ones repeat, why, and how that connects to the standard algorithm for division.
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| Geometric Discovery Record Sheet |
This depth of knowledge enables students to strengthen their number sense and build critical thinking skills within the science of mathematics. This encourages students to reach mathematical practice standards while addressing mathematical content standards.
8.14.2013
Frayer Model Sample
The Frayer Model was developed by Dorothy Frayer and her colleagues at the University of Wisconsin. This graphic organizer will lead students to a thorough understanding of new words. The corners generally include definition, facts or characteristics, examples and non-examples.
This is the modified template used in my math classroom. Based on the content, the example corner may include non-examples as well.
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In order for students to communicate mathematically, they need a deep understanding of the content. Summarizing critical vocabulary via the Frayer Model can jumpstart this process.
This vocabulary graphic organizer highlights the Common Core State Standard 7.SP.C.5 that is included in MATH-7.
This is the modified template used in my math classroom. Based on the content, the example corner may include non-examples as well.
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7.24.2013
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