This morning Gail is talking about thinking about fractions as unit fractions. The information here is from the work done by H. Wu out of Berkley. Fraction instruction starts on a number line – not with a pie diagram or other physical model. She is showing us an Nspire document that is counting by 1 fourths. The idea is multiples of unit fractions. So in the app when you are at 2 on the number line, the app shows “10/5″ as well as “ten 1/5″. We have ten copies of 1/5.
Equality is defined as having the same place on the number line. When we start with pie diagrams we are creating misconceptions. For one – we wind up not being able to be greater than one. Also we need a different circle for each fraction type.
Gail takes a minute to tell us that words in the common core matter much more than they ever did before. When the core talks about “unit” it really means it.
Next we talk about unit squares. How could we prove 15/6 = 5/2. A number line model or a unit square model will work. Now think about a unit square with 3/4 of it filled in. That is your unit. Then a half filled in unit square would be 2/3. A unit square split into 8 rectangles with 3 filled in has a value of 1/2. One and one half filled in unit squares has a value of 2. You can see these examples here.
On to addition: – we think about additional as copies of unit fractions. Multiplication is sort of a traditional area model.
Gail started by playing a little game with us. On her TI-Nspire software she drew a triangle and measured one of the sides . Then she locked the “area” and then moved one of the points. She asked “What do you think is going to happen?” Ultimately: What does a trace of all the places you can move one of the points look like? (Try it!)
Next game is what happens if we instead lock the perimeter and trace? What do you think will happen? (Try it – your hint is that it has something to do with the sum of two of the sides of the triangle.)
Gail’s point is that the only reason we ask questions is to find out what students are thinking. To probe or to push their thinking. If we aren’t doing that with our questioning, then our questions aren’t worth it. Tasks that push and probe should be our focus. A possible task – Which glider goes farther: one that starts at 25 meters high and goes 185 meters horizontally, or one that starts at 25 meters high and goes 155 meters horizontally. (One audience member asked which distance) This is an example of different context for the same concept. In terms of the Common Core this type of task hits a lot a the Math Practices.
Next: Draw triangle ABC. Construct the perpendicular bisector of sides AB and BC. Make a conjecture about the perpendicular bisector of AC. Move point A. What do you observe? Cool – but how can we make this problem more interesting? October recipes from No More Cookbook Math (Harper and Edwards 2011) has a rubric for thinking about Teacher Centered vs Student Centered.
Stein 2000 talk about choosing solutions, sequencing solutions, managing solution strategies, ask questions, consolidate the math using student work. Gail would call this “purposeful walking”. The work on formative assessment says that hands should never be up unless the student is asking you a question. You do the planning and choosing.
Say you poll the class on a particular problem (maybe TI Navigator style) what do you do with disparate results? You could have a representative from each solution justify their responses…or what about just asking the whole class what the people who answered A. were thinking, and what were the people thinking that chose whatever. Or what about pairing kids up with the goal of convincing each other that they are right. Then re-poll the class. Now you can deal with the outlying kids one on one.
Next we looked at the Nspire activity What_is_a_Solution. A simple and brilliant activity.
Lastly we did a sequence activity with the Fibonacci sequence and I’ll try and recreate it here if I can find time. In the mean time – here are Gail’s slides from the presentation:
In her last session, Gail mentioned using technology as a tool. We need to help students:
Confront their misconceptions.
Structure their knowledge.
Identify and clarify key concepts.
Play with ideas.
Students learn if:
They are actively involved in choosing and evaluating strategies.
They explore contrasting cases and notice differences.
To get students to do these things, Interactive Math Boxes on the TI-Nspire are a great way to start. The first thing we do is create a set and then evaluate a function using those set values:
The discussion then will center around how we know that the two sets that came up equal are actually algebraically equivalent. We would chalk-talk a little about what we could do to prove equivalence. Going back to the Nspire then we can have the students find another expression that is equivalent and the Nspire will give instant feedback. So pick a misconception (in the above example we were getting at the idea that some kids might add 3+4 first before distributing) and build a math box to confront it.
For the next example we made a slider and entered an expression:
Can I make it zero? Can I put in a y to make 22? Can I make an equivalent expression? These are all things you might do with the slider model.
Next we used an interactive math box to roll some dice:
In the last example we looked at, we considered how weird it would be to walk into a Starbucks and find that 31 out of the 52 people there were female:
One final video showing the same example above but using the randbin command:
Always nice to see Gail! As always, she started with a problem: How many handshakes are possible with 3 people? With 5? Then find a general rule for n people. Some of us drew a picture, others made a list. One smartie-pants worked out n pick 2. Gail mentioned the triangular numbers and talked about how this can lead in to the idea of discrete vs continuous and quadratic functions. A simple problem that can lead to some rich discussions. She’ll revisit the triangular numbers at the end of the session.
Gail emphasized two major ideas that may be new to us in the Common Core Standards:
Words matter. You must read the standard and implement the thinking and reasoning behind it.
Statistics for all students. For example, making inferences, or developing a margin of error, or deciding if differences between parameters are significant.
Make sense of problems and persevere in solving them. For example, a 5 oz steak is so much and an 8 oz steak is so much, how much for a 20 oz steak? We need to have a discussion with students about their solutions and go deeper. Do students make a decision based on just one of the data points, or do they use both? If they find a curve of best fit, what do the coefficients and constants mean in terms of the problem?
Construct viable arguments and critique the reasoning of others. So, make conjectures and evaluate arguments. For example, here is a pentagon. Make another pentagon with the same area. Students then look at six examples and decide which is correct and why. Something like area in the common core needs to be taught by the end of sixth grade and not taught again. It will be built upon and used in later grades but not taught again.
Use appropriate tools strategically. Technologies like Wallwisher or Linoit and Google Docs can be used to get kids involved when they are not at school. Gail also suggested that we think about why we use the technologies we do in class. Is it always necessary to use your interactive whiteboard? Does using it imply a front and center approach to teaching?
Reason abstractly and quantitatively. Think about a list of passing leaders in the NFL. This is kind of a Dan Meyer thing. Especially after you show the formula for quarterback passing ranking. What questions does this raise? (In Gail’s list the # of attempts for Fran Tarkenten was missing, btw.) Who’s the best? Why does the formula add 50? Why are some ratio’s weighted differently than others? Think about all the relationships that students could try and make sense of, for example, make a scatter plot of touchdowns vs. interceptions thrown. What does it look like and why?
Model with mathematics Think about the number of spaces from Go and the cost of the properties on a Monopoly board. Put a moveable line on the data and discuss what the slope means. Think about alternate scenarios like what if the board was a pentagon, or how much would Community Chest cost? In a high school setting, look at the regression line and notice that the line is pulled off course by the railroads. Maybe then we write two separate relationships: one for the railroads, and one for the rest of the properties. Henry Pollak has a book coming out from COMAP that will be awesome for everything you want to know about modeling.
Attend to precision Do we say reduce or simplify? Is there a difference? Do we use expression when its really an equation? Do we define our variables properties? Pay attention to equivalence!
Look for and make use of structure. 14/(2x-5) = 7/(3x+4) What do you do. Multiply the RHS by 2 right? Would our kids do that? Why not? We must be doing something wrong. Maybe we don’t choose our problems carefully enough…
Look for regularity in repeated reasoning Choose two whole numbers. Compute a^2 + b^2, then a^2 – b^2, then 2ab. What do you observe about the results? (they turn out to be sides of right triangles). Look at a spreadsheet where a goes from 2 on and b goes from 1 on. What do you notice? I’ll give you a hint: look for the triangular numbers. We want kids to see that math comes up in a lot of interesting places, and there may be unexpected connections.
In conclusion:
Think deeply about simple things. -Ross
Never say (do) anything a kid can say (do) -Reinhart
If the class ends after the students have explained their work, there is no need for a teacher. -Takahashi
When students don’t seem to understand something, my instinct is to consider how I can explain more clearly. A better way is to think “They can figure this out. I just need the right question.” -Kennedy
I know what they have learned when I observe them in a place they have never been -Cuoco
