Root Beer or Cola? During dinner at a local restaurant, the six people sitting at Table A and the eleven people sitting at Table B ordered the drinks shown below. Later, the waitress was heard referring to one of the groups as the “root beer drinkers.” To which table was she referring? Leah: One thing they may say: “Table A has twice as many root beer drinkers than cola drinkers.” Table A Table B

Think-Ink-Pair-Share Share your answer and thinking with a neighbor. What strategies are you using to decide which group are the ‘root beer drinkers?’ How are your thoughts alike and how are they different? Share your ideas in our math congress! Maria: “math congress” terminology is used by Fosnot and Dolk

Two Perspectives on Thinking Absolute Thinking (uses additive strategy) Comparing the actual number of root beer bottles from Table A to Table B. How might an additive thinker answer which is the root beer table? How might they justify their reasoning? Relative Thinking (uses multiplicative strategy) Comparing amount of root beers to the total amount of beverages for each table. How might a relative thinker respond to this task? Maria: To help distinguish use of terms – absolute and relative refer to type of thinking, additive and multiplicative refer to strategies, and how to solve a problem based on the type of thinking used. However, many texts/readings use the terms interchangeably, which is fine. Table A Table B

Do Snakes Grow the Same Way? Two snakes, one starting at a length of 4 feet and the other at a length of six feet grow to be 8 and 10 feet long respectively. How much did they change? Use 2 different representations to ‘show us, convince us’ why your answer(s) are correct. ---adapted Lamon (2012), pp.40-41. Working on multiplicative thinking: 1st snake doubles his original length and 2nd snake is (how many times) the original, 1 and 2/3. (this is on the next slide) Ask to model with Unifix cubes if no one uses this representation.

The Two Perspectives In absolute terms both snakes changed the same amount. Using an additive strategy, they each are longer by 4 feet. In relative terms, the first snake has grown twice as long, while the second has only grown 1 and 2/3 as long. Using a multiplicative strategy, the change is much larger for the smaller snake. The question “How much did they change?” is not specific in the type of change being requested!

Absolute and Relative Comparisons in Real Life? Can you think of some situations where it might be more useful to make an absolute comparison? Identify and list at least 2 situations. Are there other situations where it might be more useful to make a relative comparison? -Speed limit (going 10 mph over vs. going 20 mph over) (Examples are on the next slide) Cell phone and TV plans Hourly versus annual salary

Some Examples You might use absolute comparisons when looking at annual salaries, but a relative comparison when looking at per-hour wages. An absolute comparison might tell you that extended cable TV is more expensive than basic cable, but a relative comparison might tell you that you receive more channels per dollar on extended cable. A truck may be able to travel further than a compact car before needing to be refueled (an absolute comparison), but the compact car may travel more miles per gallon (a relative comparison).

Moving from Additive to Multiplicative Strategies This is a challenging, yet key transition for students. Compare these two polygon trains:

“How many?” vs. “How much of?” Example 1: Julie plans to take 3 vacation days the week of the 4th of July. How many days will she work during this holiday week? How much of the week will she work? Example 2: The number of students attending a ballet class increased from 12 to 15. How many additional students joined ballet? How much of an increase was the change? Asking questions this way makes the difference between an absolute change (“how many?”) and relative change (“how much of?”) explicit! By working through examples such as these, students begin to understand that relative change does not give an amount of change but a “times” bigger or smaller (a scale or percent of change like three sevenths of a week or 25% more students). For example 1, the ambiguity of a 5-day or 7-day work week is useful for discussion, shows there can be more than one correct answer, and that’s fine.

Which is the better investment? Think-Ink-Pair-Share Brittany: Invest $300  Receive $600 Christina: Invest $500  Receive $800 Absolute: $300, they are the same Relative: Brittany’s return is 100% of her investment. Christina’s return is 3/5 or 60% of her investment.

Unpacking the Perspectives More Absolute thinking Additive strategy Use in situations involving: counting, adding, joining, subtracting, separating, removing Relative thinking Multiplicative strategy Use in situations involving: shrinking, enlarging, scaling, duplicating, exponentiating, fair sharing, using rates

A Bit of a Stretch: Multiplicative Comparison Create and write down one question about the worms below that uses an additive comparison. Next, create and write down a different question using a multiplicative comparison. Can you find a ratio for the multiplicative comparison? Leah: “You might write the ratio of the length of worm A to the length of worm B as 11/2:1, 11/2 to 1, or simply 11/2. You could also report equivalent ratios, such as 3:2, 3 to 2, or 3/2, as well as 6:4, 6 to 4, or 6/4. You could express the ratio of the lengths of worm B to worm A as 2:3, 2 to 3, or 2/3, as well as 4:6, 4 to 6, or 4/6, in addition to 1:11/2 or 1 to 1 1/2.” From EU, p. 18