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Effective Math Instruction 6-8

December 18, 2012Teachers are thus free to provide students with whatever tools and knowledge their professional judgment and experience identify as most helpful for meeting the goals set out in the Standards.~ Introduction to the CCSSLearning OutcomesDescribe the overview of 6-8 math curriculumIdentify properties of the RDW modeling technique for application problemsDescribe and apply tape diagramsEvaluate foundational and challenging problems from gr. 6-8

AgendaState Overview of scope and sequence and modulesTape Diagram ProblemsGrade Level ProblemsAssessment Problems (PARCC)

Describe the overview of 6-8 math curriculum

4Structures in curriculumMath ModulesSequence of Modules

Module Details

Tape Diagram Problems Tape diagrams are best used to model ratios when the two quantities have the same units.

Tape Diagrams: Q1 1. David and Jason have marbles in a ratio of 2:3. Together, they have a total of 35 marbles. How many marbles does each boy have?

Tape diagrams are visual models that use rectangles to represent the parts of a ratio. Since they are a visual model, drawing them requires attention to detail in the setup. In this problem David and Jason have numbers of marbles in a ratio of 2:3. This ratio is modeled here by drawing 2 rectangles to represent Davids portion, and 3 rectangles to represent Jasons portion. The rectangles are uniform in size and lined up, e.g., on the left hand side, for easy visual reference. The large bracket on the right denotes the total number of marbles David and Jason have (35). It is clear visually that the boys have 5 rectangles worth of marbles and that the total number of marbles is 35. This information will be used to solve the problem. 5 rectangles = 35 marbles Dividing both numbers of rectangles and marbles by 5 1 rectangle = 7 marbles This information will be used to solve the problem. David has 2 rectangles and 2 x 7 = 14 marbles. Therefore David has 14 marbles. Jason has 3 rectangles and 3 x 7 = 21 marbles. Therefore Jason has 21 marbles. 10Tape Diagrams : Q22. The ratio of boys to girls in the class is 5:7. There are 36 children in the class. How many more girls than boys are there in the class?

This problem is set up like the previous one. The reasoning will also begin in the same way. 12 rectangles = 36 children By dividing both numbers of rectangles and children by 12 1 rectangle = 3 children Using the visual model above and the rectangle comparison, this problem could be solved the same way as the one above. This would be done by figuring out the number of boys and the number of girls and then subtracting. However, using the visual model, there is a second approach. When looking at the model, it is clear that the girls have two more rectangles or parts than the boys. Since 1 rectangle = 3 children then 2 rectangles = 6 children. There are 6 more girls than boys in the class 11Tape Diagrams Q3: Comparing 3 itemsLisa, Megan and Mary were paid $120 for babysitting in a ratio of 2: 3: 5. How much less did Lisa make than Mary?

10 rectangles = $120 1 rectangle = $12 Since Lisa made 3 rectangles less than Mary, she made $36 less than Mary.

12Tape Diagrams Q4: Different RatiosThe ratio of Patricks M & Ms to Evans is 2: 1 and the ratio of Evans M & Ms to Michaels is 4: 5. Find the ratio of Patricks M & Ms to Michaels.

This problem begins with comparing Patrick to Evan. Patrick gets two rectangles (parts) to Evans one. Next Evan is compared to Michael. The Length of Evans rectangle cannot change, but this time it is broken into 4 smaller rectangles. Michaels five rectangles are drawn the same size as Evans. Now it is possible to break Patricks rectangles into pieces that can be compared to Michaels. Since Evans original rectangle is equal to four of the smaller rectangles, then it follows that each of Patricks rectangles is equal to 4 of the smaller rectangles. Using that comparison, Patrick has 2 large rectangles, which can be each broken into 4 smaller rectangles, giving him a total of 8 smaller rectangles. Since the smaller rectangles are the same size as Michaels, they can be compared. The ratio of Patricks M & Ms to Michaels is 8 : 5. 13Tape Diagrams Q5: Changing RatiosThe ratio of Abbys money to Daniels is 2: 9. Daniel has $45. If Daniel gives Abby $15, what will be the new ratio of Abbys money to Daniels?

As in previous problems, the problem starts with the same setup. Since the ratio of Abbys money to Daniels is 2: 9, Abby is given 2 rectangles, while Daniel is given 9. The problem states that Daniel has $45, which is broken into 9 rectangles. 9 rectangles = $45 1 rectangle = $5 Daniel gives Abby $15, which is equal to 3 rectangles. Now the comparison looks like this. Abby Daniel Based on the new model, it is possible to determine the current ratio of Abbys money to Daniels money. The ratio of Abbys money to Daniels is 5 : 6. It would be possible to extend the question further. Students could determine how much Abby had at the beginning, and how much they each have now. The tape diagram lends itself to illustrating/modeling many different types of problems. 14Double Number Line

Double number line diagrams are best used when the quantities have different units. Double number line diagrams can help make visible that there are many, even infinitely many, pairs of numbers in the same ratioincluding those with rational number entries. As in tables, unit rates (R) appear in the pair (R, 1). Double Number Line: Finding average rate It took Megan 2 hours to complete 3 pages of math homework. Assuming she works at a constant rate, if she works for 8 hours, how many pages of math homework will she complete? What is the average rate at which she works?

To solve this problem it makes sense to use a double number line rather than a tape diagram; the double number line lends itself to the comparison of quantities in two different units, e.g., pages and hours. To complete a double number line like the one above, first label the pages on the top line. Starting with zero, mark the top line with multiples of 3, since Megan completes 3 pages in every time interval. On the bottom line label hours, this time going by multiples of 2 since the time interval is 2 hours. It is now possible to answer the first part of the question if she works for 8 hours, how many pages of math will she complete? Since the number of hours is given and the number of pages is the unknown, it is necessary to solve the problem by looking at the bottom number line, which is in hours. On the bottom number line, find and label the time of 8 hours. The corresponding number of pages is on the upper number line and connected to the 8 hour mark; in this case it is 12. So, at this rate, Megan will complete 12 pages in 8 hours. The second part of the question requires a slightly different approach to extract information from the double number line. The question what is the average rate at which she works? is asking for a unit rate. Here is an example of unit rate: if one travels 240 miles in 4 hours, then the unit rate is 60 miles per hour or 60 miles for each hour. Unit rates involve the expression: Per something which means for each something. In this particular problem the unit rate is in the form of number of pages per hour or number of pages for each hour. This means the time interval associated with the unit rate is 1 hour. Since 1 hour is not already on the double number line, it needs to be located and added to the bottom number line. Therefore the space between 0 and 2 on the bottom number line is divided in half and the label of 1 (for 1 hour) is added. Since 1 hour is halfway between 0 and 2 on the hour number line, the corresponding number of pages will be half way between 0 and 3 on the page number line. Halfway from 0 to 3 is 1 . This is labeled on the page number line. It is now possible to answer the question What is the average rate at which she works?. The answer is 1 pages per hour. 16Identify properties of the RDW modeling technique for application problemsRead (2x)Draw a modelWrite an equation or number sentenceWrite and answer statementUnitObjectContextUse RDW to solve ProblemModelling Challenge 2 boxes of salt and a box of sugar cost $6.60. A box of salt is $1.20 less than a box of sugar. What is the cost of a box of sugar?SaltSaltSugar$1.20$6.603 parts = $6.60- $1.203 parts = $5.401 part = $5.40 3 = $1.80$1.20+$1.80= $3.00Challenging ProblemsThe students in Mr. Hills class played games at recess. 6 boys played soccer 4 girls played soccer 2 boys jumped rope 8 girls jumped rope

1) Compare the number of boys who played soccer and jumped rope using the difference. Write your answer as a sentence as Mika did.

2) Compare the number of boys who played soccer and jumped rope using a ratio. Write your answer as a sentence as Chaska did.

3) Compare the number of girls who played soccer to the number of boys who played soccer using a ratio. Write your answer as a sentence as Chaska did.

Mika Said: Four more girls jumped rope than played soccer.Chaska Said: For every girl that played soccer, two girls jumped rope.Mr Hill Said: Mika compared girls by looking at the difference and Chaska compared the girls using a ratioChallenging ProblemsCompare these fractions:

Which one is bigger than the other? Why?Grade Level Problems

Using Grade level packets, explain the exemplar solution of problems.Wrap upThanks for coming!Linkswww.btboces2.org/mathpdhttp://www.parcconline.org/samples/mathematics/grade-6-slider-rulerhttp://www.parcconline.org/samples/mathematics/grade-7-mathematics

www.Engageny.org