Learning Math: Patterns, Functions, and Algebra
Solving Equations Homework
Session 6, Homework
In the puzzles that follow, each shape stands for a number value of “weight.” To solve these puzzles, make the weights in each part of the mobile balance from left to right, just as a sculptor might balance all the parts of a mobile. Here are the rules:
- The right and left sides of each horizontal beam must balance.
- Each shape has a unique and consistent weight within a puzzle, and no shapes weigh 0.
- Take the clues at face value. For example, if a clue says that the square’s weight is a multiple of the triangle’s weight, you can assume that the triangle does not weigh 1.
- All weights are either one- or two-digit, positive whole numbers.
- A piece hanging directly below the fulcrum does not affect the balance between the left and right arms. Although this piece has its own definite weight, it remains “neutral” for the purpose of balancing the other two arms.
- The size of the pieces has no relation to weight.
- These mobiles are exercises in balancing number values. They do not take into account the distance from the fulcrum.
Tip: Think about the total weight of each branch of the mobile, which may give you the values of certain weights or help you find equations relating weights. In tougher puzzles, make a list of possible values for a weight, then try to reduce that list to one correct value. Problem H4 is a system of equations with eight variables, and is very difficult.
Discover the value of each of the shapes. The total weight is 36. All shapes weigh less than 10.
Discover the value of each of the shapes. The total weight is 80. Only one shape weighs more than 9.
Discover the value of each of the shapes. The total weight is 54. The three arms are equal in weight.
Discover the value of each of the shapes. The total weight is 180. Each of the three arms is equal in weight.
Marcus had some cookies. He wanted to give them all away. He gave one half of them to his friend David. They divided the remaining cookies evenly among David’s three brothers, so each got 4. How many cookies did Marcus have originally?
Tip: Which of the methods of solving equations that you have learned in this session can be useful here? There is more than one answer.
Balance problems excerpted from In the Balance, by Lou Kroner (New York: Creative Publications, Wright Group/McGraw-Hill, 2000).
The above materials may not be reproduced without written permission of Creative Publications.
Stacey, Kaye. “Ideas about Symbolism That Students Bring to Algebra.”
Reproduced with permission from The Mathematics Teacher, © 1997 by the National Council of Teachers of Mathematics. All rights reserved.
The circles must each be 9, because the mobile splits twice. The clue tells us the cubes cannot weigh more than 5 each. Because the hourglass and 2 cubes weigh 18 and the hourglass weighs less than 10, each cube is 5 and the hourglass is 8.
The upside-down cone is 10 (2 splits), and the circle is 5 (3 splits). Because the cube and the diamond make 5 together, the clue tells us the diamond is 2, and the cube is 3. The others: hourglass = 7, cylinder = 6, horizontal bar = 1, pyramid = 8, flat disk = 9.
The flat disk must be 9 (2 splits). The cube must be an even number, and larger than the circle — this makes it 8. The others: circle = 5, pyramid = 3, cylinder = 2, diamond = 1.
One possible course is to find the value of the diamond first. After the diamond in the third set, there are three consecutive splits (to the cylinders), so the total weight after the diamond must be a multiple of 8. This limits the possible values of the diamond shape to 4 or 12, because the diamond on the left, when paired with a circle, weighs a total of 15. Investigating the circles in the left branch shows that the circle must be 11, and the diamond must be 4. The others: cylinder = 7, flat disk = 8, pyramid = 14, hourglass = 6, cube = 3, upside-down cone = 10.
Backtrack or work from false position.
Backtracking: Each brother got 4 cookies, so the brothers shared 12. This is half of the original number of cookies, so the original number is 24 (multiply 12 by 2).
False position: Pick a convenient number of cookies for Marcus, then see what would have happened. Say that Marcus had 60 cookies (a good number, since 60 is very divisible). Half go to David, so there are 30 to split among 3 brothers. If Marcus had 60, each brother gets 10 cookies.
So, to turn 10 into 4 we need to multiply by 0.4, which shows that Marcus had (60)(0.4) = 24 cookies to start.
As with Problem B10, finding the solution by covering up is possible, but difficult. A solution found by doing the same thing to both sides is only possible after creating an equation for the entire situation.
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.