About the Math Center

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Lewis Clark State college houses the Idaho Regional Mathematics Center for Region 2. The center is directed by Kacey Diemert and supported by Ryan Dent, our Regional Mathematics Specialist. The intent of the center is to provide professional mathematics support with both content and pedagogy to K-12 teachers in Region 2. The members of the Regional Mathematics Centers have experience in K-16 mathematics education, designing and delivering professional development, instructional technologies, and educational research. We are able to provide both regional and school-specific support in mathematics education. We welcome input from schools and districts as to the type of professional development they need. Our professional development begins with promoting mathematical thinking, problem solving, and the habits of mind students need to effectively understand and apply mathematics.

Wednesday, December 14, 2016

Mapping Lesson

Architects in Action

Show students a map of the United States and point out the scale in the map key. Explain that sometimes we shrink objects or make them larger so they are easier to work with. The map is a scale model of an object that is too large to represent on paper. Other scale models represent objects that are too small, such as a diagram of an atom or a magnified view of a computer chip. Review the scale on the map. For example, the scale may say that 1 inch is equal to 50 miles. Explain that a scale is a ratio used to determine the size of a model of a real object. In this case, the map of the United States is the model.
Illustrate how to draw an object to scale. Use a ruler to draw a square on the board with sides that equal 10 inches in length. Ask students how they might use this square to draw another that is half its size. Now measure and draw a second square with 5-inch sides. Explain that when an object is scaled down, the length of its sides must be reduced by the same amount. Compare the corresponding sides of the two squares. The ratio of the small square to the larger is 5: 10. Show that a ratio can be expressed in three ways: 5: 10, 5 to 10, or 5/10, which is a fraction that reduces to 1/2.

  1. Explain that students will use ratio to make a scale drawing on graph paper of the classroom floor plan including the desks, tables, closets, and so on. Divide students into teams of four. The class may choose to use either metric or English measurements.
  2. Construct a class data table on the board with three columns labeled "object," "measurement," and "scaled measurement." Students should copy this table in their notebooks and fill in the answers as they measure the objects.  Once teams have recorded all their data, they will decide on the scale of their floor plan. 
  3. Students can determine their scaled equivalents by cross-multiplying. Students should recall that when both sides of an equation are multiplied by the same amount, the equation remains balanced. 
  4. Have students use their scaled measurement, rulers, and graph paper to draw the floor plan their team measured. Remind them to include a title, labels, and a scale.
  5. As students complete their drawings, have them calculate the perimeter and area of their classrooms. What is the relationship between the drawing and the actual classroom? They should notice that the ratio of these areas is the square of the scale they chose. For example, if a scale of 0.5 inch = 1 foot was used, the ratio of areas of the drawing to the actual room will be (0.5 inches)2 = (1 foot)2 or 0.25 square inches = 1 square foot.

Additional Exercises 

  • Compare your classroom floor plan to that of another student. How are they similar and different? Which would be more useful to a construction worker trying to build a classroom in a new school? Why?
  • List other instances in which you use ratio to compare objects in your daily life. Why is it important to maintain the same scale for each measurement you record when making your model?
  • Debate the merits of using the metric system and the English system to measure lengths. Explain how to convert between the two systems.
  • Using what you have learned about ratios, proportions, and scale models, create four word problems for other students in your class to solve. For example: A square carpet measures 8 feet x 4 feet. Suppose the scale of a drawing containing the carpet is 1 foot to 1/4 inch. What are the dimensions of the carpet in the drawing? The answer: 2 inches x 1 inch.

Monday, December 12, 2016

Wordless Geometry Conundrum

From NCTM's September 2016 issue of Mathematics Teacher

What fractional part of the regular pentagram is shaded?

Given Answer: 1/2
A regular pentagram can be decomposed into triangular regions as colored in the figure below. The yellow and turquoise triangles are congruent. The problem's shaded region consists of two turquoise triangles, one yellow triangle and one red triangle, as does the unshaded region.

Friday, December 9, 2016

Glimpse into Lesson Studies in the region

5th Grade Lesson Study – Introductory Lesson on Division with Decimals
December 7, 2016

Research Question:

What are the essential understandings that students need when dividing decimals, and what can we do to ensure all students learn something new towards 5NBT.7?

By providing students with opportunities to start the problem-solving process on their own without a teacher-prescribed strategy, represent their thinking, share their thinking with their peers, and listen to their peers’ way of thinking, students’ current understandings are illuminated in a way that drives decision-making throughout the lesson.  In this lesson, 19 out of 23 students were able to provide some representation of their thinking and/or problem-solving strategy on a concept that they’ve had no previous experience with. 

Developing expertise in estimating with decimals, paired with students’ reasoning behind their estimates, prior to experiences using computation with decimals strongly supports understanding.  Estimation supports reasoning, elicits evidence of students’ place value understanding and relative size of decimals, which were found to be 2 of 3 key ingredients in developing understanding of decimal computations (per van de Walle).  Given significant evidence that students rarely refer back to their estimate after determining a more precise answer (in this lesson, and based on the experiences of all teachers in the group), there is a need to support students in increasing their buy-in of estimation as a tool for problem-solving.  Also important to initially developing understanding of division with decimals, understanding the relationship between multiplication and division can serve as an entry point for students, and simultaneously supports understanding of both models of division (partitive and quotitive).

A third essential understanding of dividing with decimals lies in the concept of equivalence.  It seemed that in working towards developing generalizable methods for dividing with decimals, equivalence will become increasingly important.  Understanding that 1.1 could also be called 11 tenths will support the generalized method of re-writing the expression that involves decimals (3.6 divided by 0.4) to work with an equivalent expression that does not (36 divided by 4). 

Therefore, there seems to be a potential instructional sequence that involves significant work with estimation of decimal values to determine relative size of decimals (individually and with computation), place value understanding and equivalence, with equivalence having the closest connection to the more common generalizable methods for dividing with decimals.  What is more certain is that introducing procedures or algorithms, such as the long-division algorithm, before these understandings are in place can inhibit students’ reasoning, intuitive thinking and repertoire of problem-solving strategies.

Ideas for further study…

Why are students not transferring use of strategies/models to new situations?

How do we know when students are making meaning of models in a way that they will choose to use it in another situation?

Friday, December 2, 2016

In the Library

What's Math Got To Do With It?

A recent assessment of mathematics performance around the world ranked the United States thirty-sixth out of sixty-four countries in the study. When the level of spending was taken into account, we sank to the very bottom of the list. We are falling rapidly behind the rest of the developed world when it comes to math education- and the consequences are dire.

In this straight forward and inspiring book, Jo Boaler named by the BBC as one of eight people who "are changing the future of education," outlines concrete solutions that can transform students' math experiences, including classroom approaches, essential strategies for students, and advice for parent. Now updated to address the controversial Common Core, this is a must read for anyone who is interested in the future of our children and our country!

Contact the Regional Math Center if you would like to borrow this or any of our other resources