Learning Math: Patterns, Functions, and Algebra
Proportional Reasoning Homework
Session 4, Homework
Sandy made some iced tea from a mix, using 12 tablespoons of mix and 20 cups of water. Chris and Pat thought it tasted great, but they needed 30 cups of tea for their party. Lee arrived, and they found they disagreed about how to make 30 cups that tasted just the same:
|Chris||It’s easy: Just add 10 tablespoons of tea and 10 cups of water. Increase everything by 10.|
|Pat||Wait a minute. 30 is just 1 and a 1/2 times 20, so since you add 1/2 as much water, add 1/2 the tea: add 10 cups of water and 6 tablespoons of tea.|
|Sandy||I think about it this way: We used 12 tablespoons for 20 cups, so 12/20 = 3/5 tablespoons for 1 cup, so for 30 cups we should use 30 x 3/5 = 18 tablespoons.|
|Lee||Wait: 20 – 12 = 8, so you want to keep the difference between water and tea at 8. Since there are 30 cups of water, we should use 30 – 8 = 22 tablespoons of tea. That will keep everything the same.|
Critique each of these methods. Which methods are the same? Which methods will really produce tea that tastes the same?
Tip: Which are absolute comparisons, and which are relative ones? What type of comparison is more useful here?
- Draw a right triangle with legs 3 and 4 cm and hypotenuse 5 cm.
- Draw a triangle whose legs are double those in H2(a).
- Draw a triangle whose side lengths are each 2 cm more than those of the first triangle.
That is, the lengths are 5 cm, 6 cm, and 7 cm.
- Which of the new triangles looks similar to the original triangle?
Five brothers ran a race. The twins began at the starting line. Their older brother began behind the starting line, and their two younger brothers began at different distances ahead of the starting line. Each boy ran at a fairly uniform speed. Here are the rules for the relationship between distance (d meters) from the starting line and time (t seconds) for each boy.
|Adam||d = 6t|
|Brett||d = 4t + 7|
|Caleb||d = 5t + 4|
|David||d = 5t|
|Eric||d = 7t – 5|
- Which brothers are the twins? How do you know?
- Which brother is the oldest? How do you know?
- For each brother, describe how far from the starting line he began the race and how fast he ran.
Tip: Which number in each equation is related to that brother’s speed? A table may help you to see the relationship.
- Which line below represents which brother? What events match the intersection points of the lines?
- What is the order of the brothers 2 seconds after the race began?
- Which two brothers stay the same distance apart throughout the race?
How do you know, based on their graphs?
How do you know, based on their equations?
- If the finish line was 30 meters from the starting line, who won?
- Which brothers’ relationships between distance from the starting line and time are proportional? How do you know?
Problem H3 taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.331-332. www.glencoe.com/sec/math
Chris’s method will produce tea that has too much mix. This method would work only if the original recipe called for the same amount of tea and water, which it doesn’t. In terms of fractions, 22/30 is not the same proportion as 12/20.
- Pat’s method is correct: 6/10 is the same proportion as 12/20, so the new mixture will have the right proportion of mix.
- Sandy’s method is also correct: 18/30 is the same proportion as 12/20.
- Lee’s method is identical to Chris’s method. If you keep the difference between tea and water the same, there will be too much tea mix added at 30 cups.
- Chris and Lee’s suggestions are incorrect because they are absolute comparisons, where a relative comparison is needed to keep the proportion of tea mix the same.
d. The triangle with side lengths 6, 8, and 10 is similar looking. A good comparison is the measures of the three angles of the triangle.
- The twins are Adam and David. We know they start at the starting line, so they must be the ones without any constants in their equations.
- The oldest must be Eric, since he is the one whose equation includes the instruction “-5”, which means he begins 5 meters behind the starting line.
- Adam started at the starting line, and ran at 6 meters per second. Brett started 7 meters ahead, and ran at 4 meters per second.
Caleb started 4 meters ahead, and ran at 5 meters per second.
David started at the starting line, and ran at 5 meters per second.
Eric started 5 meters behind the starting line, and ran at 7 meters per second.
- The points of intersection on the graph represent when one brother passes another during the race. Their times (on the horizontal axis) and their distance from the start (on the vertical axis) are the same.
- Use t = 2 in all five equations, or refer to the graph. The order is: Brett (15 m), Caleb (14), Adam (12), David (10), and Eric (9).
- Caleb and David, who run at the same speed. We know this because their graphs form parallel lines, which always stay the same distance apart.
- Find the finishing times for the five by solving the equations for d = 30, or referring to the graph (draw a horizontal line at distance d = 30). Solving the equations shows that Adam and Eric tie for first (5 seconds), then Caleb (5.2), Brett (5.75), and David (6 seconds).
- Adam and David. Their graphs pass through the origin (0, 0) and their equations are in the form y = kx.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.