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__Using Multiple Representations to Think About Middle
School Math__

Karen Gartland

- Doing this task on ratios using manipulatives gives access to kids who
don’t know the procedures for comparing ratios

- Make bar diagrams online with Thinking Blocks

- Patternary game (from her book Well Played 6-8):

- Given the
first two steps of a visual pattern

- The first
team draws a possible next step

- The
second team guesses a possible next step

- If the second team matches the
first team, the round is over. If the second team does not match the first
team, the first team shows what they drew for the next step in the pattern.
Then they repeat the process for the next step until the two teams match/are
thinking of the same pattern.

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__Multiple Representations on the Number Line through the K-8 Standards__

- In order to emphasize understanding of measurement and the
unit on a number line, have students construct a number line on a piece of
adding machine tape using the cuisinare rod unit block (or some other length
measure) to measure and mark the intervals

- With a number line, students need to understand…

- what is the unit
- the numbers on the line increase to the right and decrease to the left
- the number line represents distance
- distance between 0 and 1 is “unit distance”
- unit distance can be used to locate additional points both inside and outside the unit interval
- once two numbers are marked on a # line, all other locations are fixed

- open number line tasks as formative assessment

- Example 1 (Goal: how do kids do with relative size of intervals, are they inclined toward -#s and fractions): Label an open # line with a tickmark of 2, put some tickmarks to the left and right and label
- Example 2 (Goal: what is the comfort level with fractions, do they understand the concept of inbetween on a # line) Label -1, 0, and 2 on the number line. Where would you place \(\frac{1}{2}, \frac{3}{2}, \frac{-1}{2} \frac{7}{8}\)

- Construct right triangles in order to exactly place
irrational numbers on a number line (See this article from NCTM’s Mathematics
Teacher )

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__What If…__

Steve Yurek

- Towers of Hanoi

- Traditionally (see this geogebraapplet) there are 3 posts, and you are trying to predict the minimum number of moves for any number of discs
- What if there are 4 posts?

- Red and yellow chips: You have a pile of chips that are
red on one side and yellow on the other. Some number of chips, r, have the red
side facing up. Blindfolded, split them into to groups. Then make it so that
the number of red chips is equal in the two groups. What is the strategy that
will make this work every time?

- Solution: Count out r chips that will be in one group. Flip over all the chips in that group. You will now have an equal number of red chips in each group.
- At this point, my question was WHY? I have figured it out since then, but I want to figure out more ways to set my students up to have that burning need to figure out why.

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__Multiple Representations Make Learning Come Alive in
Elementary and Middle School__

Anne Collins

- Conjecture board: Record student conceptions before
introducing a new topic, then have students see if they can disprove the
conjectures through a counter-example (it only takes one)

- Example: What do you get when you multiply any two numbers?

- Which elementary models build into algebra?

- Partial products (and partial
sums) extend better into algebra than the standard algorithm

- Using the algeblocks quadrant
map for multiplication and the algeblocks basic model map for integer addition
and subtraction—directionality shows the sign (just like the # line) rather
than the color of the chip

- Ratios on the Cartesian plane: comparing \(\frac{2}{3}\)
and \(\frac{3}{5}\)

- \(\frac{2}{3}\) is steeper than \(\frac{3}{5}\)so by inspection \(\frac{2}{3}>\frac{3}{5}\)
- Can convert these fractions into percepts by letting each interval on the y-axis equal 10%, then look at the points (10, ?)
- Anything under y=x is going to be a proper fraction or ratio
- Vertical lines show common denominator
- Horizontal lines show common numerator
- Subtract the fractions by finding the distance between the lines (it’s not constant on the graph because the x value determines the denominator)
- For division: \(\frac{2}{3} \div \frac{3}{5}\), can think about how many \(\frac{3}{5}\) fit into \(\frac{2}{3}\). If you look at x=15, you can see that \(1\frac{1}{9}\) \(\frac{3}{5}\)s fit into \(\frac{2}{3}\)

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__Structure Through Skip Counting: Seasonal Activities to
Keep Kids Counting__

Susie Schneider and Sarah Clark

- Skip Counting Progression: Put finished projects in a
triangle with 1 in the first row, 2 in the second row, 3 in the third row, etc.

- Start with chart paper and recording it all together
- Students do the skip counting on the own paper, but can still go up to the triangle to count

- Other skills (using glue, tape, scissors, working on
handwriting, planning, sharing) progression

- Step-by-step as a whole class
- Show the whole process, then students do it themselves
- Directions written on the board, only show highlights

- Book suggestions

- “Mathematics Their Way” Mary Baratt-Lorton
- Number Wonders: 171 Activities toMeet Math Standards & Inspire Students, Grades K-2 by Catherine Jones Kuhn

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__Preparing Students for High School Algebra: Critical
Foundations for Developing Function Sense__

Judy Curran Buck

- Function sense indicators

- Students can translate between different representations
- Appreciation for and ability to apply the function concept and procedures to a real world setting

- Foundations for developing function sense

- Number sense: flexibility with numbers, and recognizing characteristics and relationships between numbers
- Operation sense: meaning of operations, relationships of operations, when to use operations
- Symbol/variable sense: generalize a pattern, use an unknown, joint variation
- Expression sense: being able to translate between English and math, equivalent forms, properties, equal sign
- Graph sense: coordinate plane, describe a relationship between two variables and make predictions

- Data collection ideas for lines of fit: jumping jacks (x =
time), the wave or wrist squeeze (x = # of people), # of pennies in a cup to
break the noodles (x = number of noodles)

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__No More Rainbows, Butterflies, Or…__

- Main point: “Get away from the cutesy and empower the
students to use appropriate language”

- We need high expectations for all kids around appropriate math language and representations. It is an equity issue that students with IEPs and ELLs are most often taught these tricks

- Needs in math education:

- Move away from computations without understanding
- Shift from getting the answer to understanding the problem
- Use errors to explore student misconceptions

- A focus on key words leads to kids not recognizing that
this problem is not possible to solve

- Visual approximation of square roots. Ex: \(\sqrt{31}\)

- Have students construct a circle: Start with a point, draw
line segments of equal length through it so that the point splits the segment
in half.

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