 Early Childhood through Eighth Grade in the heart of Silicon Valley

# Teaching & Learning Blog

Tanya Spishak

During the months of October and November, students explore a multitude of applications for the concept of proportional reasoning in their math lessons (i.e. the ability to make comparisons and identify relationships between objects, quantities, etc.). Our department experts share this window into the development of proportionality throughout the Woodland math program at each grade level.

Kindergarten

In Kindergarten, students prepared for Halloween by connecting to their math learning. Students explored and predicted which objects would weigh more or less than a pumpkin. During the pumpkin investigation, students compared the weight of a pumpkin to many other objects including an iron, milk carton, glue, and pencil. Before weighing the objects, students explored them, holding each in their hands, then made predictions about which object would be heavier or lighter. Students then used a balance scale to compare the two objects by looking to see which side of the scale went down and which side went up.

Developing Proportional Reasoning In Grade 1: November Number Corner

This month’s workouts delve into fractions and telling time. Students discover a pattern of friendly animals chomping snacks into wholes, halves, and fourths in the Calendar Grid, and the focus on fractions carries over to Calendar Collector. We introduce telling time to the hour, and students use fractions of a circle to consider whole and half on an analog clock.

We work with students to devise sentences that involve different forms of each fraction as we record them on the chart. For example, we might record, “Each dog got half of a cookie” the first time the dogs appear. The next time we might write, “Each dog ate one of the sandwich halves.” The first time the mice appear, we might record, “Each mouse nibbled on a fourth of the cookie.”  And the next time, “The sandwich was cut into quarters, and each mouse got one.”

While students need to know the names of the fractional parts, they also need to hear the language of the fractional parts being counted. For example, calendar marker 5 shows the cookie divided into four equal parts. While each part is labeled as 1/4 , the students should hear the parts counted as “one-fourth, two-fourths, three-fourths, four-fourths; four fourths make a whole!”

Developing Proportional Reasoning In Grade 2: Using Rectangular Arrays

The Daily Rectangle in second grade is a yearlong exploration of rectangular arrays and equations. In October’s Number Corner, students worked together to make as many different rectangular arrays as possible with quantities of tiles to match the day’s date.  For example, on the fourth day of the month they discover that they can push 4 tiles together to form two different rectangular arrays – 1 row of 4 or 2 rows of 2. As the month progressed, students found that they could form more rectangular arrays with some quantities than with others.  Students practiced daily recording of addition equations to represent their arrays.

Towards the end of Unit 2 in October, students extended their exploration of arrays to a lesson entitled “Thinking About Two’s”.  In this lesson, students brainstormed a list of things that come in 2’s. They selected one of the ideas – nostrils in this case– by class vote.  Then, each student drew a picture of the selected idea and it was then mounted on a class chart to illustrate a pattern of counting by 2’s.  The lesson continued for two sessions, as the students discussed observations about the pattern that was created.  The 2’s chart is used again to extend counting by 2’s through 100 on a hundreds grid and more observations were shared about the resulting patterns found.

While equal groups of objects are often used to introduce multiplication, rectangular arrays provide a more powerful and flexible way to model and solve basic multiplication combinations, while setting the stage for multiplication in third grade.

October’s Number Corner was designed to give students who may already be familiar with cups and quarts a sense of the size and measure of the two metric units of liquid volume—liters and milliliters. The activities provided ongoing experience with fractions, mixed numbers, and multi-digit computation. Students were able to watch a collection that grows by one-fourth of a liter each school day, and made estimates earlier in the month to predict how many liters would be filled by the end. Because they were tracking the total number of milliliters at the same time, they had an ongoing opportunity to add multi-digit numbers as well as fractions and mixed numbers.    Students begin working with egg carton models in 3rd grade.  This model allows for visualizing fractions both as part of a whole or parts of a set.  Thus 8 out of 12 eggs can be viewed as ⅔ of the whole, or 4/6.  Yarn or string is used to flexibly divide the model into sets/portions to deepen understanding.

In our second unit in 5th Grade Level math, Adding and Subtracting Fractions, students use ratio tables and proportional reasoning as an introduction to finding common denominators.

Until now, students have worked with ratio tables where the ratio is 1 to something, a unit rate. For example, 1 ball costs \$15. In this unit, students begin with non-unit rates: 5 pounds of granola costs \$6, and learn to find the unit rate; students then use that information to find other rates.

Students also begin to realize that they do not have to find the unit rate to find many other equivalent rates. For example, to find the price for 10 pounds, students can double the 5 pounds for \$6 to get 10 pounds for \$12.

Our goal is that students begin to generalize which operations work in ratio tables and can look to the relationships in the numbers in the problem to help them decide what to do in a flexible, efficient way.

These strategies and models are extended to solve a variety of story problems and to make generalizations about finding common denominators. At the end of our unit, students gain more explicit experience with greatest common factors and least common multiples as they find common denominators and learn to simplify fractions. Ever play billiard ball or pool? If you could perfectly hit a ball from a corner at a 45-degree angle, the ball trajectory could look like the ones above. The sizes of the five different tables above share a common feature: length is double the width. Back in August and September, 5th-grade students investigated that if the dimension of the tables is proportionally the same, then the trajectories of the ball for each table would also be the same. Another example is shown at the right.

Pre-Algebra

Our second unit of study in Pre-Algebra builds the big idea of proportional relationships in the context of real-world situations. Students look at similar figures in order to determine how the figures change when the sides are enlarged or reduced uniformly. Then students investigate how to use and make scale drawings.  Students extend their ideas of proportional relationships, comparing relationships in the context of bank accounts, bulk food purchases and gas mileage in cars. They also revisit a million-penny tower problem from earlier in the course. Students examine tables, graphs, and situations to identify characteristics of proportional relationships. At this stage, students use informal strategies to solve proportional situations.  Later in the year, students will connect their understanding of scaling and scale factors to other proportional relationships such as distance, rate and time, percents, and interest.

Algebra

In Algebra, proportional reasoning is often investigated with graphs and pictures. An example is the enlargement and reduction of pictures. If the distance of eyes is doubled, the length of the sword should also be doubled. The orange line in the graph indicates the scaled results (if one thing is doubled, the rest should also be doubled). If a point is above the orange line, then the feature is enlarged too much. If a point is below the orange line, then that feature is reduced too much.

Geometry

Proportional reasoning is a central theme in Geometry.  It is most commonly used in our unit on similarity and dilations.  It also comes up in one method we use to prove the Pythagorean Theorem.  Students further explore relationships between perimeter, area, surface area and volume of similar objects.