Chapter 1: TE Equations and Functions
Overview
Equations and Functions consists of eight lessons that introduce students to the language of algebra.
Suggested Pacing:
- Variable Expressions - \begin{align*}1\;\mathrm {hr}\end{align*}
- Order of Operations - \begin{align*}1 \;\mathrm{hr}\end{align*}
- Patterns and Equations - \begin{align*}1-2 \;\mathrm{hrs}\end{align*}
- Equations and Inequalities - \begin{align*}1-2 \;\mathrm{hrs}\end{align*}
- Functions as Rules and Tables - \begin{align*}0.5 \;\mathrm{hrs}\end{align*}
- Functions as Graphs - \begin{align*}1 \;\mathrm{hr}\end{align*}
- Problem-Solving Plan - \begin{align*}0.5 \;\mathrm{hr}\end{align*}
- Problem-Solving Strategies: - \begin{align*}2 \;\mathrm{hrs}\end{align*}
- Make a Table; Look for a Pattern
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Problem-Solving Strand for Mathematics
The problem-solving strategies presented in this chapter, Make a Table and Look for Patterns, are foundational techniques. Making a Table can help structure a student’s ability to organize and clarify the data presented and teach the student to communicate more clearly. When teaching Look for a Pattern, ask questions such as:
- Can you identify a pattern in the given examples that would let you extend the data?
- Do you observe any pattern that applies to all the given examples?
- Can you find a relationship or operation(s) that would allow you to predict another term?
Alignment with the NCTM Process Standards
Two promising practices, focused on the communication standards, can be effectively used with these strategies. The first is setting aside a time on a regular basis (i.e. once a week) to have students write about the problem solving they have been doing (COM.2). Present a daily warm-up problem which is well suited to the strategy being learned such as Look for a Pattern. Students work on one problem a day, first individually and then as a group with teacher leadership and summation. Each day the problem is discussed and solved, and many different points of view are shared in the process (COM.1, COM.3). At the end of the week, students are asked to write about any one of the problems they did earlier in the week. This practice pushes students to develop logical thinking skills, to benefit from the classroom work that was shared earlier in the week, and to learn to communicate mathematical ideas clearly (COM.4). It also gives students a choice; they only have to write about one of the problems, and it can be the problem that made the most sense to them. Oftentimes, unfortunately, we do not honor enough what makes sense to students; we expect them only to be able to follow the logic presented to them. We must give them experiences of “sense-making” as well (RP.1, RP.2).
A second practice is posting student work, such as:
- gallery walks where work in progress can be viewed
- posters that highlight exceptionally well done solutions
- student essays posted on the classroom wall
What matters is that students’ work is valued and shared (RP.3). If these pieces are large enough to be seen from a distance, this practice helps students to recall various approaches to solving problems (RP.4) and keeps their thinking “alive.”
- COM.1 - Organize and consolidate their mathematical thinking through communication.
- COM.2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
- COM.3 - Analyze and evaluate the mathematical thinking and strategies of others.
- COM.4 - Use the language of mathematics to express mathematical ideas precisely.
- RP.1 - Recognize reasoning and proof as fundamental aspects of mathematics.
- RP.2 - Make and investigate mathematical conjectures.
- RP.3 - Develop and evaluate mathematical arguments and proofs.
- RP.4 - Select and use various types of reasoning and methods of proof.
- 1.1.
Variable Expressions
- 1.2.
Order of Operations
- 1.3.
Patterns and Equations
- 1.4.
Equations and Inequalities
- 1.5.
Functions as Rules and Tables
- 1.6.
Functions as Graphs
- 1.7.
Problem-Solving Plan
- 1.8.
Problem-Solving Strategies: Make a Table; Look for a Pattern