Pattern Generalizations Goals 1. Teachers helping children develop algebraic thinking related to function. 2. Teachers learn how to ask questions related to three algebraic habits of mind. 3. Teachers help children use justification to develop algebraic thinking.

Ratio and Proportion Review “the ability to reason proportionally develops in students throughout grades 5-8. It is of such great importance that it merits what ever time and effort that must be expended to assure its careful development” (NCTM, Curriculum and Evaluation Standards, p. 82)

Ratio and Proportion Review What did you learn most from the ratio and proportion baseline and summative assessments? Describe at least two ways that you helped your students think proportionally. Describe one classroom situation where you saw a student exhibit growth in thinking multiplicatively.

My Number Patterns 1, 3, 5, 7, ____, ____, ____ … 1, 2, 4, 8, ____, ____, ____ … 31, 28, 31, 30, 31, 30, 31, ____, ____, ____ … 27, 49, 79, ____, ____, ____ … 8, 5, 4, 9, 1, 7, 6, 3, 2 eight, five, four, nine, one, seven, six, three, two, zero

Border Problem Figure out without talking, without writing, and without counting one by one how many unit squares are on the border of the 10 by 10 square.

What were they thinking? Sharmeen: 4 x 10 – 4 Colin: Joe: Melissa: (10 x 10) – (8 x 8) Rita: 4 x 9 Zach: 4 x 8 + 4

Grid Problem There are 15 rods in the figure below. Draw the figure on your paper. Write an arithmetic expression that describes how you drew your picture.

Toothpick Problem

Toothpick Analysis explicitrecursive tabular non-tabular

Baseline Assessment

Marci’s Dots This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work. A pattern of dots is shown below. At each step, more dots are added to the pattern. The number of dots added at each step is more than the number added in the previous step. The pattern continues infinitely. (1 st step)(2 nd step)(3 rd step) 2 dots6 dots12 dots Marcy has to determine the number of dots in the 20 th step, but she does not want to draw all 20 pictures and then count the dots. Explain or show how she could do this and give the answer that Marcy should get for the number of dots.

Students A and B

Students C and D

Students E, F, and G

Students H and I

1992 NAEP Results – Marci’s Dots

1992 NAEP Scoring Guide

Perimeter Problems