Quantitative Reasoning “You’re soaking in it!”. I agree with… If students are eventually to use algebraic notation and techniques to express their ideas.

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Presentation transcript:

Quantitative Reasoning “You’re soaking in it!”

I agree with… If students are eventually to use algebraic notation and techniques to express their ideas and reasoning productively, then their ideas and reasoning must become sufficiently sophisticated to warrant such tools. Pat Thompson, ASU

And… “In appealing to algebraic methods to solve complex problems, we elect not to use less formal approaches that are tied more closely to the situation. We also tend to devalue this “informal” reasoning in comparison to algebraic methods.”

Quantitative Reasoning is: Quantitative Reasoning is when you solve a problem by reasoning about the quantities and relationships within the problem, without the support of variable assignments, algebraic expressions, and equations. (We need an example!)

An Example Problem 1. I walk from home to school in 30 minutes, and my brother takes 40 minutes. My brother left 6 minutes before I did. In how many minutes will I overtake him? A typical algebraic solution to this problem involves assigning variables, writing algebraic expressions, and eventually stating and solving an equation. Let’s try together to write an algebraic solution.

Now solve without formal Algebra Problem 1. I walk from home to school in 30 minutes, and my brother takes 40 minutes. My brother left 6 minutes before I did. In how many minutes will I overtake him? …use a diagram, graphs, words, ideas, etc.

An example of QR I imagine myself walking behind my brother, seeing him ahead of me. What matters in catching up with him is the distance between us and how long it takes for that distance to become zero. The distance between us shrinks at a speed that is the difference of our walking speeds.

and… I take 3/4 as long as brother to walk to school, so I walk 4/3 as fast as brother. Since I walk 4/3 as fast as brother, the distance between us shrinks at the rate of 1/3 of brother’s speed.

and… The time required for the distance between us to vanish will therefore be 3 times as long as it took brother to walk it in the first place (6 minutes). Therefore, I will overtake brother in 18 minutes.

Problem 2 At some time in the future John will be 38 years old. At that time he will be three times as old as his daughter Sally. Sally is now 7 years old. How old is John now?

Problem 3 Jim, Sue, and Tom played marbles. Sue won 6 marbles from Jim and 5 from Tom. Jim won 3 marbles from Tom and 4 from Sue. Tom won 12 marbles from Jim and 2 from Sue. (Compare Tom’s number of marbles before and after the game.)

Problem 4 An elementary school has two 4th grade classes, Room 5 and 7, and two 5th grade classes, Room 6 and 8. There are 52 4th graders in the school, which is 7 more students than are in the 5th grade. There are 22 students in Room 6. How many students are in Room 8?

Problem 5 The ratio of the number of blue beads to the number of red beads in a jar was 2:5 at first. Ian removed ¼ of the blue beads and 2/5 of the red beads from the jar. Find the new ratio of the number of blue beads to the number of red beads.