ALGEBRA IN ACTION In the sixth grade's most recent investigation, the "realistic situation" has gotten a little more fictitious than the previous contexts, with frogs as the main characters. To quote the unit authors, "This unit uses the context of the famous short story by Mark Twain, "The Celebrated Jumping Frog of Calaveras County," to develop equivalence and its use in solving algebraic problems." The number line is the predominant model for this investigation, used to represent competing frogs jumping down a track. It moves naturally to a double number line, in which one frog's progress (a.k.a. one mathematical expression) is represented on the top of the line, while another frog's progress (a.k.a. the other equivalent expression) is represented on the bottom side of the same line. In this way, students grapple with determining the unknown value of each jump (the variable known as "j") if both frogs landed on the same ending point.
The Big Ideas of this investigation include:
An algebraic expression can be treated as an object (not only a procedure). This often challenges students who are comfortable with solving x + 3 = 8, but become confused with how to interpret x + 3 by itself. What does that mean?
Variation: Variables describe relationships and are not merely unknown quantities
Equivalence: Algebraic expression can appear different yet be equivalent objects
Equivalent expressions can be used interchangably
Equivalent amounts can be separated off
Equivalent expressions can be operated on by + or -, x and division, to give new equivalencies
exploring Parks and playgrounds
MULTIPLICATION AND DIVISION OF FRACTIONS In this realistic set of stories, mathematicians are asked to explore the a series of situations that requires them to extend their understanding of multiplication and division to fractional amounts. The story begins with a number line model that represents a marathon race course, the identify exact portions of the course run this year compared to last year, based on landmarks like water stations and and course markers (at every half, quarter, eighth). Next, students utilize a ratio table to analyze marathon training data, which creates some natural confusions (opportunities to learn!) with whether they are using multiplication or division when determining running rate. These "warms ups" lead into a deep look at the open array model (area model or rectangle) as students evaluate park maps, to determine the area of proposed playgrounds and black top four-square courts. By the final stages of this series, students have discovered several fantastic methods for modeling multiplication and division of fractions that can underpin the usual algorithms we are all familiar with (does anyone remember, "don't ask why, just invert and multiply"???).
Big Ideas explored in Parks and Playgrounds:
Fractions represent division
The whole matters
To maintain equivalence, the ratio of the related numbers must be kept constant
The properties (distributive, associative, commutative) that hold for whole numbers, also hold for rational numbers
The relationship between multiplication and division of fractions
Ratios, rates and best buys
Exploring Ratios, Rates, Fractions and Decimals in Realistic Situations
Sixth grade students have explored the challenges we all face every day when we make real life decisions. The first exploration looked at cat food deals offered at two competing pet food stores, to determine which offered the best deal. This required that students manipulate whole numbers, fractions and decimals, while maintaining the same ratio of cost per can. We discovered the importance of unit price in the process.
Students became pet food store employees in the second investigation, creating a cost chart for birdseed, based on weight in pounds, so customers would know the cost of various bag sizes based on a set unit price. Students continued their work as pet food store employees during investigation #3, when they helped their store owner determine how to expand her famous puppy chow recipe to different size bags. They determined fractional amounts for each ingredient that maintained the same ratio within the whole recipe, as well as the cost of each bag weight, based on the original unit price. Phew!!! The thinking was outstanding!!!
Students moved onto a new context, looking at the gas gauge of a local produce store owner, who needs help deciding if he has enough gas to drive a new produce pick-up route. At this point, these mathematicians are digging deep in order to pull together the information they need from a mishmash of given details. The farmer's gas tank is 5/8 full, they know round-trip gas tank usage, but this is a one-way loop, with a new road never driven before. They know some of the mileage but not all of the distances. They are comparing, converting and combining different fractional amounts of the gas tank (thirds, eights, twelfths), relating those fractional amounts of the gas tank to the mileage the farmer can drive per fraction (if he can drive 600 miles on a full tank, how far can he drive on 5/12 of a tank??). In the end, they have a recommendation for the farmer - should he fill up his gas tank before he leaves on his pick-up route or does he have enough gas in his tank already?
The final investigation will pull together the entire unit, when students plan a road trip to visit friends along the Pennsylvania Turnpike. How real is that??
Ideas that have emerged thus far:
Unit rate is determined through division.
In order to maintain the same ratio for different amounts, one can use multiplication and division by the same number.
In order to maintain the same ratio for different amounts, one can use a ratio table and addition of ratio parts.
Common denominators can make operations with fractions easier!
A ratio can be written a : b : c, as a relationship between parts.
A double number line can help find fractional amounts of whole numbers.
integrated art with ratios
We were lucky enough to find time for integrated art with Ms. Iannuzzi, merging mathematical ratios with the creation of colors in art. As a class, we found the ratios of color drops needed to make primary, secondary and tertiary colors on the color wheel. We scaled up those color drop ratios to make enough cookie frosting to create a cookie color wheel for every student. Check out the back of the food coloring box - maybe your student can use the ratios listed there to create his/her own Easter egg coloring or holiday cookie frosting this year!
field trips and fundraisers
Exploring Fractions in Realistic Situations
Field Trips and Fundraisers is our first investigation from the Context for Learning Mathematics curriculum resource. In this investigation, students determine a solution to a problem we all face - how to share limited food. In this case, "students" head out on a "field trip" in four separate cars. The kids in those cars have to equally share the submarine sandwiches they have received. In one car, 4 kids share 3 subs. In another car, 8 kids share 7 subs. In the other cars, 5 kids share 4 subs, and another 5 kids share 3 subs. How much of a sub does each student get? When the "students return to school," a debate ensues - did each car get the same amount to eat?
The investigation continues on, merging cars into two groups, and finally merging all the kids on to one bus. Each new investigation builds on the prior work, dividing subs into fractional parts, comparing fractions, adding and subtracting fractions.
When the submarine sandwich conundrum has been thoroughly investigated, students are presented with a new challenge - design a bike race course map for a 60 kilometer race, with a multitude of water, food and rest stations at fractional distances along the way. If there needs to be a water station at every tenth of the race, what kilometer marker is that? Where do the food stations go if they need to be placed at every fifth of the entire course?
Students are expected to use common sense and prior knowledge to solve the problem they are facing. When the investigators converge for a Math Congress, students share their work and explain their thinking as clearly as they can, while their peers actively listen, ask questions and consider the alternative strategies being presented. The teacher guides the discussion to highlight efficient strategies and important ideas, while prompting students to make sense of their own work and the work of others.
Ideas that have emerged thus far are exciting!
To add fractions together, it helps if they are divided into the same-size parts. Find common denominators certainly helps!
Using benchmark fractions like 1/2 and 1/4 can help divide up wholes into equal parts. It also can help to use benchmark fractions to determine a rough estimate of a wonky fraction's value (17/22 is about equal to 3/4).
The fractional parts each student received follow a pattern - the number of subs, divided by the number of students, results in the fraction each student receives! Fractions tell a story of division!
When the numerator is divided by the denominator, the equivalent decimal is revealed!