Ratios, Proportions, and Proportional Reasoning Chapter 18 Ratios, Proportions, and Proportional Reasoning

Ratios vs. Fractions What is the difference? Pg. 431

Three ways to think of comparison in ratios Multiplicative comparison Additive comparison Constant relationship

Multiplicative comparison Multiplicative comparison means you are comparing two things together that need to be multiplied. Multiplicative comparison questions are usually written in word problems that have this format: Statement, Statement, Question We use the two statements to determine the number sentence or equation. An equation is just like a number sentence but it includes letters.

Additive comparison Comparing Growth, Variation 1  Alignments to Content Standards: 4.OA.A Taken from Illustrative Mathematics Task There are two snakes at the zoo, Jewel and Clyde. Jewel was six feet and Clyde was eight feet. A year later Jewel was eight feet and Clyde was 10 feet. Which one grew more? Solution Viewing this additively, both snakes grew 2 feet and therefore grew the same amount. Viewing it multiplicatively, Jewel grew 26 its length, while Clyde grew 28 its length. From this perspective, Jewel grew more. Given the purposeful phrasing of the problem, both interpretations are reasonable, but the goal is to understand the two perspectives, thus the difference between additive and multiplicative reasoning.

Reasoning for this activity IM Commentary The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking"). This would set the stage for a comparison of the two perspectives. In the case were the students don’t bring up both arguments, the teacher can introduce the missing perspective. In later grades, students will learn that "which grows more" means "which has the greater absolute increase?" and "which has the greater growth rate?" means "which has the greater increase relative to the starting amount?" but students won't see this type of language for two or three years. Teachers need to be aware of this and work to ask questions as unambiguously as possible; for example, when asking for multiplicative comparisons, use language such as, "How many times greater is x than y." They should also be prepared to address this potential for confusion along the way.

Write a multiplication equation to match each comparison statement. Activity 1: Write a multiplication equation to match each comparison statement. Write a comparison statement to match the multiplication equation. Write a multiplication equation to match each comparison statement. Comparison Statement Multiplication Equation 21 days is 3 times longer than 7 days.   8 pounds is 4 times as heavy as 2 pounds. 72 inches is 12 times the length of 6 inches. 30 fish is 5 times as many as 6 fish.

Reasoning proportionally A red car and a silver car are traveling at the same constant rate. When the red car has traveled 20 miles, the silver car has traveled 12 miles. How far will the red car be when the silver car has traveled 32 miles? A red and a silver car are traveling at different but constant rates. They pass Exit 95 at the same time. When the red car has traveled 20 miles past exit 95, the silver car has traveled 16 miles. How far will the red car be when the silver car has traveled 32 miles? Pg. 436 Then for slope use the problems on pg. 443

Battle ship Coordinate grids Coordinate grids are included in measurement and data

Developing measurement concepts Chapter 19 Developing measurement concepts

Concepts and skills: Three key elements to teaching measurement Make connections- make comparisons Use models of measurement Use actual measuring tools

estimating Develop benchmarks for comparison Chunk the lengths for comparison when estimating (smaller units of length are easier to estimate) Track your measurements using hands or other objects

Non-standard measurement Measure table side with paper Use everyday objects to measure length, width, height

Standard measurement Gummy worm activity Measure the gummy worm in inches/cm, then stretch and compare the lengths.- primary measurement

volume Gallon-man Comparing standard and metric units of measurement, using volume Solid volume- area times height 5.MD.3 Carters Candy Company

Comparing units Long Jump task 5.MD tasks in flash drive- Long Jumps

Geometric thinking and geometric concepts Chapter 20 Geometric thinking and geometric concepts

Develop geometric thinking The van hiele levels of geometric thinking Pg. 490

Geometric shapes What’s my shape? Identifying quadrilaterals My 2-d shapes book

sorts 2d and 3d sorts Attributes of each Triangles and polygons

investigations Polygon capture Triangle hierarchy Science fair project- includes coordinate grids Choose an activity

For next class For Monday- Next Wednesday- Chapters 21-23 Your mathematical philosophy Extra credit- come help me move tomorrow. (just kidding)