Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

Explain how a solution was derived and why it is reasonable. Use words, numbers, symbols, charts, graphs, tables, diagrams and/or models to explain mathematical reasoning.

A man has 3 shirts: red, black, and blue and 2 pairs of pants: jeans and slacks. Make a tree diagram to show all possible ways of combining the pants with the shirts. What is the probability that the man will wear the red shirt with the jeans?

(Sample Tree Diagram:

red

jeans black

blue

red

slacks black

blue

The probability that the man will wear the jeans and red shirt is 1/6. This can be shown using the tree diagram. In the tree diagram above, there are 6 possible choices, however only 1 out of the 6 are a red shirt with jeans.)

(2)

Danny spun a spinner two times. Each section of the spinner had the letters s-p-i-n. What is the probability that Danny will get the letter P and the letter I in any order? Use a table to explain your mathematical reasoning.

S

P

I

N

S

S, S

S, P

S, I

S, N

P

P, S

P, P

P, I

P, N

I

I, S

I, P

I, I

I, N

N

N, S

N, P

N, I

N, N

(The table shows that there are two chances to spin P and I together; P,I or I,P. This is out of a total of 16 chances. So, Danny has 2 out of 16 chances to spin P and I in any order. This is 1 out of 8 changes if you put it simplest form.)

(3)

A 4 x 4 in. rectangle has an area of 16 sq. inches. A 4 x 5 in. rectangle has an area of 20 square inches. A 4 x 6 in. rectangle has an area of 24 square inches. Make a table using the information from the problem. Identify the pattern. Then use the pattern to find the area of a 4 x 9 in. rectangle. Explain your reasoning.

(Pattern – count by fours

4 x 4

4 x 5

4 x 6

4 x 9

16

20

24

36

The table shows that each time an inch is increased by 1, the area increases by 4. To get from 4 x 6 to 4 x 9, the area will have to increase 3 times. 3 x 4 = 12. This means the area will increase by 12. 24 + 12 = 36. The area of a 4 x 9 inch rectangle is 36 sq. inches. )

(4)

Two quarts of pink lemonade will serve 5 people. At that rate, how many quarts will Nina need to serve 30 guests? Explain how you found your answer.

(To find the answer, you need to understand that 30 people are 6 times more people than 5. Next, you use this information to make the quarts 6 times more. 2 quarts x 6 quarts = 12 quarts. Nina will need 12 quarts of lemonade to serve 30 guests.)

(5)

Draw a diagram of use numbers to show a gain or loss for: loss 2 points, gain 4 points, loss 3 points, gain 5 points, gain 2 points, loss 9 points.

(-2 + 4 = -2; -2 -3 = -5; -5 + 5 = 0; 0 + 2 = 2; 2 – 9 = -7 points, This shows a loss of 7 points.)

Mathematical reasoning is needed for problem solving. In general, sixth grade students should be able to problem solve by using 3 phases of a problem solving process. These phases allow children to think thorough a problem logically to determine a solution. The first phase of problem solving is to be sure to read and understand the problem. The second phase is to plan and solve the problem. The last phase is to look back at the problem and check work. The more children practice working through each of these phases, they better they will become at using mathematical reasoning to problem solve.

In addition, children must be able to explain their mathematic reasoning; this includes how an answer was obtained and why it is reasonable. This can be done with words or in writing. Children can use visuals such as: tables, charts, diagrams and tables to both obtain a solutions and explain their reasoning. Kinesthetic learners will benefit from making models. These can be 3-d representations that are created or found. For example, if your child is asked to find the number of corners on a rectangular prism, he/she could use a shoe box as the model.

(2)

Tree diagrams are one way to organize all possible outcomes for compound events or name specific probabilities. They are a very helpful visual to be used to show mathematical reasoning. Below is an example of how a tree diagram can be used to show all possibilities for tossing a coin (heads or tails) and a number cube (1-6).

It is easiest to use the least number of possibilities as your start point. Since the coin only has two possibilities: heads or tails, we will start here.

Heads

Tails

Next, we will think about the outcome of all possible outcomes for tossing the number cube. We will add all these possibilities to both heads and tails.

1

2

Heads 3

4

5

6

1

2

Tails 3

4

5

6

We can use the tree diagram to find all possibilities by counting all possibilities shown when the diagram is completed. In this case, there are 12 possible combinations of the coin and the number cube.

A tree diagram can also be used to find a specific theoretical probability. For example, if we needed to find the probability of getting tails and a 3, we would look at the tree diagram for tails and find how many 3s there are. There is only one 3 with tails, which means one possible outcome. To find the sample space we will need to use the number of all possible combinations, 12. So, we have a 1/12 chance of getting tails and 3.

Visual learners may want to draw heads and tails instead of writing the words. They can also show the dots or numbers on the number cube.

Kinesthetic learners will best be able to create the tree diagrams if they have the actual experiment times to look at, hold, and investigate. For example, you can have them use a spinner from a board game and a die. They will then be able to create a tree diagram for each part of the spinner with each dot on the die.

(3)

A grid can be helpful in showing all possible outcomes for repeating an event two times. The numbers, letters, or word going across the top represent all possible outcomes, as do the numbers on the far left side. The remaining pairs of numbers, letters or words show possible outcomes after both events have occurred. A problem where grid could be helpful would be: Danny spun a spinner two times. Each section of the spinner had the letters s-p-i-n. What is the probability that Danny will get the letter s on both spins?

To create your grid, you need to determine how many rows and columns you will need. There are 4 letters on the spinner, but you will need 5 rows and 5 columns in your grid so that you have room to show the letters across the top of the grid and going along the side from top to bottom. Begin your grid by setting this up, as seen below.

S

P

I

N

S

P

I

N

Next, fill in the middle of the chart by first listing the letter shown on the row on the left and the column above. An example is shown below.

S

P

I

N

S

P

P, I

I

N

The P was listed first, because the P is the row, I is second because this is the column. The same pattern should be used to fill in the remaining grid items. (You could choose to list the letters vertically by column, but you would just need to keep this consistent in all columns. Choose a pattern and stick with it.)

S

P

I

N

S

S, S

S, P

S, I

S, N

P

P, S

P, P

P, I

P, N

I

I, S

I, P

I, I

I, N

N

N, S

N, P

N, I

N, N

The visual learner may need to use color-coding to keep the actual possibilities separate from the events. This can be done by highlighting all the pairs or using a different colored pencil for the original event.

S

P

I

N

S

S, S

S, P

S, I

S, N

P

P, S

P, P

P, I

P, N

I

I, S

I, P

I, I

I, N

N

N, S

N, P

N, I

N, N

Now the grid can be used to find the probability requested. If we need to find the probability that Danny can get an “s” on both spins, we need to find that on the chart as S, S. There is one S,S out of a total of 16 pairs. Thus, the probability of S,S is 1/16.

You can have your child practice creating a grid for a die that is tossed 2 times.

(4)

Creating a table or list is another way to show all possible outcomes for a compound event. Tables and lists are especially helpful when you’re asked to find a specific probability. When creating a list or inputting possible outcomes into a table it is most important to list all possible outcomes in an organized way so that no events are left off or repeated. This is why you should work on all possible outcomes one event at a time. Here’s an example. A store sells red and blue hats in sizes small, medium, and large. If Tony randomly chooses a hat from the rack without looking, what are the chances it will be a red hat, size medium? This problem is asking for the specific probability of choosing a red, medium hat. You will not be able to find this out until you know the sample space, or the total possible outcomes. All possibilities are shown in the list below.

RedBlue

Sm. Sm.

Med. Med.

Lg. Lg.

The total possibilities could also be shown in a table like the one below.

Red

small

medium

large

Blue

small

medium

large

Either organization visual will give you the answers you need. It is important that each learner decide for him/herself which method is easiest to comprehend. Both visuals show that there are a total of 6 possible outcomes. We are asked to find the probability of choosing a red hat, size medium. We can see by looking at the list or table that there is one choice of a hat that is medium and red. Red, medium is 1 choice out of 6 possible choices. So, Tony has the theoretical probability of 1/6 chances of choosing a red, medium hat.

Lists can be written vertically like the list above or horizontally. The table is horizontal and could be written vertically. It is up to each individual to create different kinds of lists and tables to determine what will work best for them.

Place different items from your home in a pile and have your child create neat lists and/or tables to show all possible outcomes for closing their eyes and choosing that item.

(5)

Having the ability to sequence and prioritize information in a problem will help children to gain more success in math solutions. The best way to put information in order or priority is by making a list or table. Practice this strategy with your child. You can try placing information in a sequenced list or using a grid to record possible data. An example of each is shown for sample problem #2.

Monica, Christine, Tracy, David and Rick are all wearing different colored shirts (red, yellow, green, pink, and white). Tracy and Christine never wear primary colors. Christine is not wearing pink. The boys are not wearing white. None of the girls are wearing green. Only David is wearing a color that has the letter Y in it. Monica is not wearing pink or white. Find the color of shirt that each person is wearing.

Make a List to Prioritize Information

red: not Christine, not Tracy, Monica is left here.

yellow: not Christine, not Tracy, is David

green: not Christine, not Tracy, must be Rick

pink: not Christine, not Monica, must be Tracy

white: not David, not Rick, not Monica, has to be Christine

Make a Table to Prioritize Information

red

yellow

green

pink

white

Monica

yes

no

no

no

no

Christine

no

no

no

no

yes

Tracy

no

no

no

yes

no

David

no

yes

no

no

no

Rick

no

no

yes

no

no

(6)

Encourage your child to explain the steps he/she used to solve a problem and explain why his/her answer is reasonable. This can be done by having your child review all three phases of the problem solving process with you.

Work with your child to help him/her successfully complete phase one of the problem solving process. Be sure that your child reads the problem a few times before attempting to solve it. In phase one, read and understand, sixth graders should be sure to identify and determine each of the following:

What exactly is being asked?

Children should underline the sentence or command in the problem and write a problem goal in their own words.

What do I know from the problem?

Children should highlight important, relevant facts that will help them to meet his/her goal from question one.

What do I know from personal knowledge?

When necessary, children will apply personal knowledge to add important facts to a problem.

In phase two of the problem solving process, 6^{th} graders will use the information obtained in phase one to evaluate relationships and determine how to solve the problem. After choosing a strategy to solve a problem, children will use that strategy to find a solution. There are several strategies that can be used to solve problems. Help your child to use the list of all the options below to determine which could be used to reasonably solve each problem he/she encounters.

Problem Solving Strategies

Choose an operation: + , - , x,

Make an organized list

Make a table

Draw a picture

Make a graph

Look for a pattern

Guess and test

Write an equation

Work Backward

Solve a simpler problem

Act it out or use objects

Be sure that your child understands that phase 3 of the problem solving process is just as important and the first two phases. Many children get an answer and stop there. Going back to the original problem, re-evaluating the question and problem facts and comparing them to the solution will allow students to use mathematical reasoning to determine if his/her answer is reasonable. On phase 3, children should:

A restaurant has a kid’s meal with the choice of a main course of a hamburger, hotdog or chicken strips. The meal comes with a side of fruit or fries. The drink choices include: milk or juice.

Draw a tree diagram and use it to explain the chances Mike has of his mom ordering him a hamburger with fries and juice.

(Sample tree diagram:

milk

fruit

hamburger juice

milk

fries

juice

milk

fruit

hotdog juice

milk

fries

juice

milk

fruit

chicken juice

strips

milk

fries

juice

Mike has a 1 in 12 chance that his mom will order him a hamburger with fries and juice. There is only one choice on the tree diagram that shows this out of 12 possible choices.)

(2)

Use a grid to show all possible outcomes if your spin a 4 section spinner numbered 1, 2, 3, and 4, two times. Use the grid to find the probability that you will spin the combination of a 2 on the first spin and then a 4 on the second spin. Use the table to explain your mathematical reasoning.

(Sample table:

1

2

3

4

1

1,1

1,2

1,3

1,4

2

2,1

2,2

2,3

2,4

3

3,1

3,2

3,3

3,4

4

4,1

4,2

4,3

4,4

The probability of spinning a 2 on the first spin and a 4 on the second spin is 1/16. The table shows one option where a 2 is spun, followed by a 4. This is out of 16 possible choices.)

(3)

Stephanie a spinner with 4 sections colored: red, blue, green and orange. Create a grid to show all possibilities if Stephanie spins two times. Use the grid to explain Stephanie’s chances of only spinning orange or red.

(Sample grid:

red

blue

green

orange

red

r,r

b, r

g, r

o, r

blue

r,b

b, b

g, b

o, b

green

r,g

b,g

g, g

o, g

orange

r,o

b, o

g, o

o, o

The grid shows that there are 4 possible choices where Stephanie would only have the chances of spinning red or orange; r,r; o,r, r,o, o,o. There are 16 total possible choices. This means, Stephanie has 4 out of 16 chances of only getting red or orange. This is equivalent to ¼. )

(4)

The table shows the number of days that Don walks each day.

Day

1

2

3

4

5

Miles

1

3

5

7

9

Use the table to explain how far Don will walk after 10 days?

(The table shows a pattern of an increase of 2 miles per day. This means that by day 6, he would walk 11 miles, 7-13, 8-15, 9-17, and on the tenth day he would walk 19 miles.)

(5)

Chris reads for 2 minutes the first day of school. After that, he doubles his reading time each day. Make a table to show how many minutes Chris will be reading by the 5^{th} day of school.

(Sample Table:

1

2

3

4

5

2

4

8

16

32

The table shows Chris’ time doubling each day. This means, by day 5, Chris will be reading 32 minutes.)

(6)

The Ross family is saving money for a family vacation. The opened a savings account with $600.00 and then began to deposit money each week. The first week they deposited $20. Each week after that, they deposited $5 more than the week before. Make a table too show how much money they will have saved by week 6. Use the table to explain how much money they will have saved.

(Sample Table:

1

2

3

4

5

6

620

645

675

710

750

795

The Ross’ will have saved $795 by the sixth week. On the first week, she had 600 + 20. From week 2 on, they deposited 5 more dollars than the week before: 25, 30, 35, 40, 45. At the end of the 6^{th} week, they had $795.)

(7)

In a seven day week, Katie practices her guitar 14 hours. At this rate, how many hours will she practice her guitar in 28 days? Explain how you found your answer?

( 7 days = 14 hours, this is two hours a day, So in 28 days, she will practice 28 x 2 = 56 hours)

(8)

Oliver has gone to the store to buy rice for his family. His father instructed him to buy the brand that is the best deal. Brand One has an 8 oz. box for $2.35 and Brand Two has a 14 oz. box for 14 oz. Which brand is a better deal? Use a visual, words or numbers to explain how you found your answer.

(Sample visual:

Brand One

Brand Two

$2.35 for 8 oz.

$4.15 for 16 oz.

29.38 cents for 1 oz.

29.64 cents for 1 oz.

The chart shows that Brand One is least expensive per oz. I found out how much it was per oz. by dividing the dollar amount by the number of ounces. Brand One is the best deal.)

(9)

Thirty percent of the 6^{th} grade class pack their lunch for school, fifteen percent have someone pack a lunch for them. What percent of the 6^{th} grade class buy their lunch? Support your answer with a Sample visual:

T he circle graph shows that more 6th graders buy lunch than pack lunch. The circle graph repesents 100%. Since blue is 30%, red is 15%, then 55% of students must buys lunch. This is because 30 + 15 + 55 = 100%.)

(10)

On a test, Mark scores 9 out of 11 on multiple choice questions, 5 out of 6 correct on true false, and 4 out of 7 on fill-in. What was Mark’s overall percent? Show the steps you use to solve the problem.

(Sample answer:

1. Add up all the total points possible for each type of question:

11 + 6 + 7 = 24

2. Add up all Mark’s scores.

9 + 5 + 4 = 18

3. Write this as a ratio of Mark’s points over total possible points.

18/24

4. Convert the fraction into a percent by dividing and moving the decimal answer two places to the right.

18/ 24 = 0.75 = 75%)

(11)

Dave is making models of solid figures. He is using gum drops as vertices and toothpicks as sides. To make a cube, he used 8 gumdrops and 12 toothpicks. How many gumdrops and toothpick will he need to make a square pyramid. Make a model, drawing, or use words to justify your answer.

(A square pyramid has a square base. Each corner of the base will need one gumdrop vertice. The top of a pyramid has just one vertice (point). So, Dave will need 5 gumdrops. Each of his four gumdrops on the bottom will need a toothpick to make the square, a four sided base. Also, each vertice on the square base will have to have a toothpick to attach to the top point. This is 4 more toothpicks. Dave will need a total of 8 toothpicks.)

(12)

Amanda the following models in front of her; a rectangular prism, triangular prism and pentagonal prism in front of her. She is using them to count the number of vertices and edges for each shape. She found that the rectangular prism has 8 vertices and 12 edges; the triangular prism has 6 vertices and 9 edges; the pentagonal prism has 10 vertices and 15 edges. Put the information in a table and use it to explain how many vertices and edges would be found in a hexagonal prism.

Triangular P.

Rectangular P.

Pentagonal P.

Hexagonal P.

6 vertices

8 vertices

10 vertices

12 vertices

9 edges

12 edges

15 edges

18 edges

(The table shows a pattern of two edges being added each time and 3 edges. So, a hexagonal prism will have 12 vertices and 18 edges.)

(13)

The Jackson family measured the perimeter of their rectangular living room and found it to be 54 feet. If the room is 15 feet in length, how wide is it? Explain, using the appropriate formula.

(The formula for the perimeter of a rectangle is P = 2l = 2 w. In this problem, P = 54 and L = 15. These numbers need to be put into the formula; 54 = 2x15 + 2 w. To find the answer, I first multiply 2 x 15 and get 30. The new problem is 54 = 30 + 2w. I needed to get the variable w, alone, so I subtracted 30 from 54. The new problem was 24 = 2 w. The last step to solve for 2 is to divide 24 by 2. So, w = 12 feet.)

(14)

Tanisha is building a circular pen to house her bunnies. She wants the diameter of the pen to be 5 feet. How much fencing should she buy? Explain, using the appropriate formula.

(Tanisha needs to find the circumference of the pen. The formula is C d. We don’t know C, but = 3.14 and d = 5. So, insert these into the formula to solve. C = 3.14 x 5. Tanisha will need 16 feet of fencing.)

(15)

A gain of $10 was followed by a loss of $4; a loss of $3; a gain of $2 and a loss of $4. This is a gain or loss of how much? Draw a diagram or use numbers to explain.

(+10 -4 = 6-3 = 3+2=5 – 4 = 1. This is a gain of $1)