Strengthening the Core: Common Core

Mathematical Practices Workshop

By Mandy Bakas

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.

CCMPS

Make sense of problems and persevere in solving them◦ Students have bought into the problem and have an interest

in it.◦ Students understand that problem can be solved by using

multiple methods and pathways (tables, graphs, diagram, equations, verbal descriptions, etc)

◦ Students use concrete objects or pictures to help solve problems

◦ Students may ask the following essential questions: Do I have a plan in place to solve the problem? Do I need to change my thinking because of alternate ways of

solving problem? Does my answer make sense?

CCMPS #1

Mr. and Mrs. Consumer make several local calls each month, but seldom do they call long distance. They are looking for a good local telephone service.

 Regional ExchangeYou can have reliable and efficient telephone service for only

$8.00 per month plus $0.16 per call.

General Telephone We offer top quality phone service for only $14 per month plus

$0.08 per call.

CCMPS #1 –Algebra I Example

Reason abstractly and quantitatively ◦ Students understand that reasoning abstractly

involves breaking down the situation symbolically◦ Students can manipulate abstract expressions

and equations based on their numeric counterpart.

◦ Students may ask the following questions: Can I translate it from English to math? Do I understand the process?

CCMPS #2

Define variables and write an equation to model the following situation:

What is the number of slices of pizza left from an 8-slice pizza after you have eaten some slices?

CCMPS #2 Algebra I Example

Construct viable arguments and critique the reasoning of others◦ Students understand that when they construct arguments

they need to be able to defend their point of view and evaluate counter-arguments.

◦ Students need to keep an open-mind.◦ Students need to be able to distinguish between a strong

argument and a weak one.◦ Students develop strategies to solve problems, question

them to understand their thinking and understanding

◦ Students may ask the following questions: Can I defend my argument? What is the value of critical listening?

CCMPS #3

Why do searches and radars use the circle concept and not any other shape such as the square, etc.?

Select a circle and any other shape, compare the two and determine which would be the most appropriate shape to be used in a search. Make sure to provide detailed support of your conclusion.

CCMPS #3 Geometry Example

Model with mathematics◦ Students understand that they can analyze problems and

apply mathematical patterns and models in everyday life. ◦ Students can interpret results and revise models for

maximum proficiency.◦ Students are able to apply the math they know to solving

problems◦ Students are able to write equations to describe a

situation.

◦ Students may ask the following questions: What tools can help me? When do I use this in my everyday life?

CCMPS #4

You have 100 acres of land to grow lettuce and peas. You want to decide how many acres of each crop to plant to make a maximum profit.

Lettuce: Investment per acre is $120, and income per acre is $150

Peas: Investment per acre is $200, and income per acre is $260

Maximum amount you can invest: $15,000.

CCMPS #4 Algebra II Example

Use appropriate tools strategically◦ Students understand that they need to choose the

appropriate tool that will allow them to explore, solve and/or deepen their understanding of the problem.

◦ Students may ask the following questions: What math tools could I use to attack the problem? What is the best method to used for this problem? What is an alternate method that can be used to

double-check or look at the problem from at different perspective?

CCMPS #5

Find all the roots for P(r) = 5x3 + 4x2 -20x - 16

CCMPS #5 Pre-Calculus Example

Attend to precision◦ Students understand that calculations must be

done precisely with careful attention to the use of appropriate units.

◦ Students do error analysis.◦ Students may ask the following questions:

Does my explanation make sense to a non-math person?

Does my claim/explanation make sense?

CCMPS #6

Interior Design: The Figueroas are planning to have new carpet installed in their guest bedroom, family room and the hallway. Find the number of square yards of carpet they should order.

Notice: the diagram is in ft and the answer needs to be in sq yds.

CCMPS #6 Geometry Example

Look for and make use of structure◦ Students understand that there are mathematical

patterns all around life. ◦ Students understand that patterns help us make

predictions.◦ Students are able to reason through problems. ◦ Students have a strong number sense.◦ Students may ask the following questions:

Is there a pattern or structure in this problem? Can this pattern or structure be used to solve and

understand the problem?

CCMPS #7

Given: circle O, M is midpoint of ABProve: OM AB  

Students realize that they need to draw auxillary lines in order to complete this proof. They need to draw in the radii AO and BO.

CCMPS #7 Geometry Example

Statement Reason

1. 1.2. 2.3. 3.4. 4.5. 5.

Looking for and expressing regularity in repeated reasoning.◦ Students understand that there are various ways

to solve similar problems and evaluate their processes to identify short cuts.

◦ Students may ask the following questions: Is there another way to solve the problem or

recognize a repeated procedure? Does it have a pattern I can use to predict?

CCMPS #8

Background: When students are learning to multiply binomials, instead of telling them the shortcuts for the special cases, have them discover it.

Discovery: Find each product. 1. (x + 8)(x + 8) 2. (y + 5)(y + 5) 3. (d – 3)(d – 3) 4. (9r – 2)(9r – 2) 5. (x + 4)(x – 4) 6. (3c + 7)(3c – 7)

Questions: Do you recognize any relationships or patterns? Can you create general rule? Using your general rule above, can you predict what the

following products would be?

CCMPS #8 Algebra I Example

Model Cycle

Model Cycle1. Identifying variables in the situation and selecting those that represent

essential features

2. Formulating a model by creating and selecting geometric, graphical, tabular, algebraic or statistical representations that describe relationships between the variables

3. Analyzing and performing operations on these relationships to draw conclusions

4. Interpreting the results of the mathematics in terms of the original situation

5. Validating the conclusions by comparing them with the situation and them either improving the model or if it is acceptable

6. Reporting on conclusions and the reasoning behind them.

Practice Algebra I Problems

Identify the CCMPs being used in each problem

Algebra I Practice #1 A motor boat traveled 12 miles with the

current, turned around and returned 12 miles against the current to its starting point. The trip with the current took 2 hours and the trip against the current took 3 hours. Find the speed of the boat and the speed of the current.

Algebra I Practice #2 Maria bought books and CDs as gifts.

Altogether she bought 12 gifts and spent $84. The books cost $6 each and the CDs cost $9 each. How many of each gift did she buy?

Algebra I Practice #3 The soccer team held a car wash and

earned $200. They charged $7 per truck and $5 per car. In how many different ways could the team have earned the $200?

Practice Geometry Problems

Identify the CCMPs being used in each problem

Link: http://commons.bcit.ca/math/examples/nucmed/algebra_geometry/index.html

Title: Nuclear Medicine: Scaling Images

Problem: In a certain imaging study the screen is set to a scale of 1:3 (i.e. 1

cm viewed on the screen is equivalent to 3 cm in the person’s body.) If a screen view of the liver displays a lesion area of 1.6 cm2, how large is the lesion area in the person’s liver?

Geometry Practice #1

Link: http://commons.bcit.ca/math/examples/chemsci/algebra_geometry/index.html

Title: Chemical Science and Geometry: Composting and Area

Problem: The composting method is to be used to remediate a contaminated site. Composting

consists of the degradation of the contaminants to simpler, nontoxic compounds using organic material. The site has the following shape.

There is a small lake on one of the borders of the site, which has an area of 0.15 km2

in common with the property. If one bag of compost can handle 3850 m2 of land area, what is the minimum number of bags required to remediate this property (lake not included)?

Geometry Practice #2

Link: http://commons.bcit.ca/math/examples/forestry/algebra_geometry/index.html

Title: Forestry: Estimating Volumes of Trees

Problem: I have a number of trees in a stand and I want to estimate the

volume of the trees. One tree in particular is 43.5m high, and the dbh (diameter at basal height) is 2.48 m. The dbh is the diameter of the tree at a height of about 1-2m above the ground. Below this point the tree trunk really spreads out and is ignored. Most of the valuable wood is in the trunk of the tree.

Geometry Practice #3

Practice Algebra II Problems

Identify the CCMPs being used in each problem

Link: http://commons.bcit.ca/math/examples/chemsci/logs_exponentials/index.html

Title: Chemical Science Logs and Exponents

Problem: Keanu Lapaloosa, a conscientious BCIT lab assistant has just heated up a solution of

sodium hydroxide, (NaOH), to 225° F as part of an experiment. He then puts the solution inside a fume hood for cooling and finds that it cools from 225° F down to 185° F in 5.0 minutes. If the temperature inside the fume hood is a constant 60° F, find, using Newton’s Law of Cooling:1. the time it takes the solution to cool from 225° F to 120° F2. the temperature of the solution after 30 minutes in the fume hood.

Newton’s Law of Cooling states: Where T is the temperature of a cooling object at time t , T0 is the temperature of the object

at time t = 0, Ts is the temperature of the surrounding medium, and k is the decay constant.

Algebra II Practice #1

Link: http://commons.bcit.ca/math/examples/chemsci/algebra_geometry/index.html

Title: Chemical Science and Geometry: Composting and Area

Problem: The composting method is to be used to remediate a contaminated site. Composting

consists of the degradation of the contaminants to simpler, nontoxic compounds using organic material. The site has the following shape.

There is a small lake on one of the borders of the site, which has an area of 0.15 km2

in common with the property. If one bag of compost can handle 3850 m2 of land area, what is the minimum number of bags required to remediate this property (lake not included)?

Algebra II Practice #2

Title: Gold – Maximum Area (parabolas)

Problem: In the 19th century, many adventurers traveled to North America to search for gold. A

man named Dan Jackson owned some land where gold had been found. Instead of digging for the gold himself, he rented plots of land to the adventurers. The “rent” was to give Dan 50% of any gold found on the plot of land. Dan gave each adventurer four stakes and a rope that was exactly 100m long. Each adventurer had to use the stakes and rope to mark off a rectangle with north-south and east-west sides. Did everyone get the same area to dig for gold? Explain your answer.

Expansion: One of the diggers discovered that one kind of rectangle always had the greatest

area. He decided to sell the secret to other diggers. What was the secret? How would you show that no other rectangle with a perimeter of 100m will have an area larger than the rectangle you discovered in the secret? Do this in two ways.

Algebra II Practice #3

Practice Pre-Calculus& Calculus Problems

Identify the CCMPs being used in each problem

Pre-Calculus Practice #1Link: http://commons.bcit.ca/math/examples/forestry/linear_algebra/index.html

Title: Forestry: Optimized Required Logging Using Systems of Equations

Problem: My logging company has a contract with a local mill to provide 1000 m³ of Lodgepole

pine, 800 m³ of spruce, and 600 m³ of Douglas fir logs per month. I have three regions available to me for logging. The following table gives the species mix, and timber density for each region.

How many hectares should I log in each operating region listed above to deliver exactly the required volume of logs? I don’t want to have to store logs so I don’t want any left over at the end of each month, but I do need to make my quota.

Region

Volume/Hectare

% Pine

% Spruce

%Fir

West 330 m3/ha 70% 20% 10%North 390 m3/ha 10% 60% 30%East 290 m3/ha 5% 20% 75%

Calculus Practice #1Problem: The controller for the EA Electronics Company has used

the production figures for the last few months to determine that the function c(x) = -9x5 + 135x3 + 10,000 approximates the cost of producing x thousands of one of their products. Find the marginal cost per unit production if they are now producing 2600 units.

Calculus Practice #2Link: http://commons.bcit.ca/math/examples/nucmed/integral_calc/index.html

Title: Nuclear Medicine: Using Integration to Determine Drug-Time Relationship in a Patient

Problem: A drug is excreted in a patient’s urine. The urine is monitored

continuously using a catheter. A patient is administered 10 mg of drug at time t = 0 which is excreted at a RATE of -3t1/2 mg/h .

1. What is the general equation for the amount of drug in the patient at time t > 0?

2. When will the patient be drug free?

Linear Programming Covering all 8 practices