Class Materials 21-51
The class materials will give a sense
of the issues we have been discussing with our students in
grades 3 and 4. Perhaps they will inspire some teachers to try
out new ideas in their own classes.
But the materials are not recipes. The
quality of classroom discussions depends greatly on the
teachers’ ability to listen, to engage students, to draw
out the implications of their statements, and to see how
particular activities fit into the grande scheme of mathematics
and the students’ evolving understanding.
Any rote enactment of lessons
(”First, do step 1, then step 2, then collect the
papers...”, etc.) is doomed to fail.
Preparation and organization certainly
help, but teaching, like playing jazz, relies on creative
improvisation and careful listening.
Lesson 26 Earning
Money
We present a problem that uses a function similar to the Dots Problem in that it uses the same numbers, but has a different story. As a class, we fill out a three-column table and write a generalized expression. Students then work on their own on a new problem.
We present a problem that uses a function similar to the Dots Problem in that it uses the same numbers, but has a different story. As a class, we fill out a three-column table and write a generalized expression. Students then work on their own on a new problem.
Lesson 27 Functions
as Patterns, Hops, Tables, and Mathematical Expressions
Three functions are represented as a sequence of patterns, a sequence of hops on the
number line, a data table, and an algebraic expression.
Three functions are represented as a sequence of patterns, a sequence of hops on the
number line, a data table, and an algebraic expression.
Lesson 28 Linear
Vs. Quadratic Functions: Focus on Differences
Two functions are represented as a sequence of patterns, a sequence of hops on the number line, a data table, and an algebraic expression.
Two functions are represented as a sequence of patterns, a sequence of hops on the number line, a data table, and an algebraic expression.
Lesson 29 Comparing
Functions
The students will discuss, represent, and solve a verbal problem involving the choice between two functions.
The students will discuss, represent, and solve a verbal problem involving the choice between two functions.
Lesson 30 Starting
with Tables
Students again work with a function, this time beginning with a table and then a formula to generate ordered pairs that follow it.
Students again work with a function, this time beginning with a table and then a formula to generate ordered pairs that follow it.
Lesson 31 Starting with a Rule
Students focus on whether given outputs are consistent with a given rule. In the initial discussion, and handout, the rule is given .
Students focus on whether given outputs are consistent with a given rule. In the initial discussion, and handout, the rule is given .
Lesson 32 Rules and Formulas
Students are given a rule and a data table supposedly generated according to the rule. Students evaluate whether: (1) the proper rule has been applied and (2) the result is correct.
Students are given a rule and a data table supposedly generated according to the rule. Students evaluate whether: (1) the proper rule has been applied and (2) the result is correct.
Lesson 33 Formulas and Stories
The students will be required to work with the relation between different mathematical expressions (formulas) and stories.
The students will be required to work with the relation between different mathematical expressions (formulas) and stories.
Lesson 34 Dinner Tables I: Separate Tables
Students work with a function relating number of tables to the number of available seats. One table seats 4, two tables seat 8, three tables seat 12….
Students work with a function relating number of tables to the number of available seats. One table seats 4, two tables seat 8, three tables seat 12….
Lesson 35 Dinner Tables II: Joined Tables
Students work with a function relating number of tables (in a straight line) to the number of available seats. One table seats 4, two tables seat 6, three tables seat 8…. The activity builds directly on the discussions from last lesson.
Students work with a function relating number of tables (in a straight line) to the number of available seats. One table seats 4, two tables seat 6, three tables seat 8…. The activity builds directly on the discussions from last lesson.
Lesson 36 Functioning Together
Students work together to develop multiple representations of a function.
Students work together to develop multiple representations of a function.
Lesson 37 Recipes
That Have Names: Times Two
The lesson centers around a function that multiplies input by two. New notations are
introduced.
The lesson centers around a function that multiplies input by two. New notations are
introduced.
Lesson 38 Recipes That (Ex)Change Ingredients
The lesson centers around a function that multiplies input by two but also changes the ingredient to another type of ingredient.
The lesson centers around a function that multiplies input by two but also changes the ingredient to another type of ingredient.
Lesson 39 The
Human Graph
Students graph the function they worked with last week for their homework, namely, k x 2 $/h. The idea is to show how the amount of pay received varies according to the number of hours worked. Give each student a large card with a place for an ordered pair: (x, y), where x refers to hours worked, and y refers to amount earned. Next class we’ll work with $3 per hour and $5 per hour.
Students graph the function they worked with last week for their homework, namely, k x 2 $/h. The idea is to show how the amount of pay received varies according to the number of hours worked. Give each student a large card with a place for an ordered pair: (x, y), where x refers to hours worked, and y refers to amount earned. Next class we’ll work with $3 per hour and $5 per hour.
Lesson 40 The
Human Graph II: Rate of Pay
Students graph the functions k x 2 $/h and k x 3 $/h. The idea is to show that for each linear function the points fall onto a straight line. Give each student a large card with a place for an ordered pair: (x, y), where x refers to hours worked, and y refers to amount earned.
Students graph the functions k x 2 $/h and k x 3 $/h. The idea is to show that for each linear function the points fall onto a straight line. Give each student a large card with a place for an ordered pair: (x, y), where x refers to hours worked, and y refers to amount earned.
Lesson 41 Rate of Pay Versus Total Pay
Students compare points on a hrs-pay Cartesian space. The main challenge lies in recognizing that, although one student earned more, the other student was paid better, that is, at a higher rate of pay. They must indicate the difference in pay and the differences in amount worked.
Students compare points on a hrs-pay Cartesian space. The main challenge lies in recognizing that, although one student earned more, the other student was paid better, that is, at a higher rate of pay. They must indicate the difference in pay and the differences in amount worked.
Lesson 42 Comparing
Graphs
Students are given an hourly rate of pay and infer coordinates for (h, $) over a range of hours. They produce a table, and a graph of work-pay. Then they produce another graph for another rate of pay and discuss differences in time and pay.
Students are given an hourly rate of pay and infer coordinates for (h, $) over a range of hours. They produce a table, and a graph of work-pay. Then they produce another graph for another rate of pay and discuss differences in time and pay.
Lesson 43 How
Many Points?
Students work with: (a) a new context—distance time; (b) generating coordinates.
Students work with: (a) a new context—distance time; (b) generating coordinates.
Lesson 44 Interpreting
Maps
A casual discussion of maps, beginning with one centered on the school, building outward to larger scales, and finally moving back into the neighborhood of the school.
A casual discussion of maps, beginning with one centered on the school, building outward to larger scales, and finally moving back into the neighborhood of the school.
Lesson 45 Interpreting
Maps II
Students construct a narrative of a trip, given a map and a table of arrival and departure times. They also determine how much time was spent along each segment of the trip (and how much time was spent at each place along the way.) If time permits, they construct a table showing ordered by time, and showing the duration of each segment and the accumulated times.
Students construct a narrative of a trip, given a map and a table of arrival and departure times. They also determine how much time was spent along each segment of the trip (and how much time was spent at each place along the way.) If time permits, they construct a table showing ordered by time, and showing the duration of each segment and the accumulated times.
Lesson 46 Maps
to Graph
Students will be given two linear distance-time graphs and asked to tell a story about each graph and to compare them. They will later explore comparisons between points in each line.
Students will be given two linear distance-time graphs and asked to tell a story about each graph and to compare them. They will later explore comparisons between points in each line.
Lesson 48 Interpreting Graphs
Students will be given two linear distance-time graphs and asked to tell a story about each graph and to compare them. They will later explore comparisons between points in each line.
Students will be given two linear distance-time graphs and asked to tell a story about each graph and to compare them. They will later explore comparisons between points in each line.
Lesson 49 Three
Heights Problem – Review
In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.
In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.
Lesson 50 The
Piggy Banks Review
The whole lesson revolves around a multipart story problem that involves changes in two quantities over several days. The initial quantities are equal yet unknown. Then transformations are applied to the quantities. Students are asked to compare the quantities throughout the week even though only their relative amounts can be determined.
The whole lesson revolves around a multipart story problem that involves changes in two quantities over several days. The initial quantities are equal yet unknown. Then transformations are applied to the quantities. Students are asked to compare the quantities throughout the week even though only their relative amounts can be determined.
Lesson 51 Dinner Tables Revisited
Students work with a function relating number of tables (in a straight line) to the number of available seats. One table seats 4, two tables seat 6, three tables seat 8…. The activity builds directly on the discussions from last lesson.
Students work with a function relating number of tables (in a straight line) to the number of available seats. One table seats 4, two tables seat 6, three tables seat 8…. The activity builds directly on the discussions from last lesson.
