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

Mathematical Reasoning: mathematical reasoning, problem solving Your child should be able to explain why he/she chose the method or description that he/she chose to illustrate his/her reasoning processes.

Ethan buries a box of candy for his sister to find. He buries it in his mother’s rose garden. Make a map and at least three clues to lead Ethan’s sister to the treasure.

(2)

You are new to town and need to find the library. Your neighbor tells you to travel North for three blocks and then turn right (East) for 2 more blocks. Make a left at the light and it’s on the right side of the street half way down the block. Make yourself a map to help you find it.

(3)

You plan a potluck to have with your nine friends. You need to make a chart of who will bring what dish. You want to make sure you have main dishes, salad, dessert, and drinks. Make a chart to help plan the potluck. Make up 9 friends and sign them up for dishes to bring.

(4)

You want to open a candy store for your friends to buy candy after school. You take a survey and find out that 5 of your friends like Skittles most, 3 friends like M&Ms and 6 friends like red licorice. Make a graph to show your friends’ favorite candy.

Decide how much you think you should order to last you the first month. Make a chart to help you order your candy.

(5)

Solve this problem. Be sure that your digits are aligned correctly: 25 x 6 (150)

It is important that your child know how to do “mental math” which is math done “in the head” without pencil/paper/finger-counting because for simple calculations you want your child to be so comfortable in manipulating the digits in numbers that he/she does not need to write down everything. Examples of mental math that a third-grader should be working toward include basic addition, subtraction, multiplication and division facts, adding two-digit numbers that do not require regrouping, multiplying numbers by 10, 100, or 1000. With this background your child is becoming proficient in the math basics needed to perform more complicated routines.

(2)

For problems that do not involve mental math exclusively, your child needs to become proficient in explaining how he/she reached the conclusion, both in writing and orally. If a more complicated problem is one that your child seems to be able to do mentally, writing down only the answer, it is possible that he/she has devised a pattern that can be used to complete the problem without further computation. Ask your child to explain how he/she reached his answer. Some children are really able to do more complicated problems and you don’t want to discourage this, but because classroom teachers cannot always listen to each child’s oral explanation, make sure your child knows how to express his/her thinking in writing and that you come to some acceptable compromise with the school so that your child is not writing out many solutions he/she can do mentally just to prove that the work is original.

(3)

A more common situation is the completion of a lesson in which the student has “memorized” how to do the work but doesn’t really understand why this approach is best. One symptom of this problem is that the child cannot “remember” how to do that type of problem after a few days or not practicing such problems. It’s sort of like being able to ride a bike; if you haven’t ridden for awhile you might be rusty and need to go slowly at the beginning, but you haven’t really forgotten how. If you notice this happening, spend more time with your child, exploring your child’s reasoning, asking leading questions that will help a child see the pattern(s) involved, and in providing manipulatives and simpler problems to use in developing an understanding of the principles involved.

(4)

Make sure that your child can write neatly and legibly all of the digits in our number system (0-9), can draw the operations symbols for addition, multiplication, subtraction, and division; these are different depending on whether problems are written horizontally or vertically.

(5)

Make sure that your child knows that word problem solutions always include a label and that several “label patterns” not only put the correct label after the numerical solution but also help you be sure that you solved a problem correctly. For example, if you want to add together the number of apples in a basket and the number of oranges, you can do that, but what will the label be? It can’t be “orange” or “apples”. Make sure that your child knows, if that really was the calculation requested (and not a mis-reading of a word problem that came with extraneous information) that the label wording can be changed to “fruit” or “apples and oranges.” If multiplying two lengths, the label will be “square (length word)” and that if adding those lengths, the label is just the name of that length, such as feet, inches, etc. If three groups of toys each contain 6 toys, then the label will be toys; there is no need for part of the label to express “group” as that is implied in multiplication problems.

(6)

Help your child learn to complete calculations in a neat fashion, with adequate room allowed on the paper to line up the digits in their place-value columns and complete the solution without running into the next problem on the page. When digits are placed casually on the page, with no regard for their place-value column, eventually your child will, for example, end up adding some numbers from a tens column with numbers in a ones column, resulting in a wrong answer when probably your child knew how to do the problem. Workbook pages and worksheets often do not allow for the amount of space that a child needs to produce legible numbers and work the problem in neat, straight columns; you may need to work with your school about how best to handle it. This caution is an example of an issue that does not greatly affect the young child so it’s easy to overlook it but will come back to haunt the learner when the problems are more complex and require most, if not all, of a page to complete. It’s easier to develop good habits now.

(7)

It is completely acceptable at the third-grade level for a student to show the reasoning on a word problem in the form of a picture or diagram rater than writing out a long explanation. The idea is to sketch the solution you plan to use as a way of helping the learner reason out what is needed for the solving of the problem and then to help remember (in the case of multi-step problems) exactly where the steps were leading. Students should not be completing some sort of a drawing later when an adult reminds them that one was expected. Teachers and parents cannot discern from that approach whether the student really understands the process and thus cannot help their growth in mathematical reasoning ability.

(8)

There are online “brain teaser” sites for children and “logic books” available at bookstores and libraries. These teach “out of the box” thinking which is very useful in helping students develop more flexible thinking patterns. Children of this age often think in very narrow terms; helping them to consider a wide range of possibilities is part of the maturation that you want to further for them. These abilities help a child to explain his/her solutions in more creative and often time-saving ways.

Try the next 7 problems to help you learn to think of creative ways to solve problems and express your ideas in a way that others can understand. Draw little pictures and diagrams to help you explain how you solved the problems.

You must cut a birthday cake into exactly eight pieces, but you're only allowed to make three straight cuts, and you can't move pieces of the cake as you cut. How can you do it?

(2)

Can you place six X's on a Tic Tac Toe board without making three-in-a-row in any direction?

(3)

Nine dots are arranged in a three by three square. Connect each of the nine dots using only four straight lines and without lifting your pen from the paper.

(4)

Arrange the numbers 1 through 9 on a tic tac toe board such that the numbers in each row, column, and diagonal add up to 15.

(5)

Arrange the numbers 1 through 9 on a tic tac toe board such that the numbers in each row, column, and diagonal add up to 15.

(6)

There are several chickens and rabbits in a cage (with no other types of animals). There are 72 heads and 200 feet inside the cage. How many chickens are there, and how many rabbits?

(7)

If a boy and a half can eat a hot dog and a half in a minute and a half, how many hot dogs can six boys eat in six minutes?

(8)

One way to write an addition problem is this way:

3 + 5 = 8. Can you write it in another format? Line up the numbers carefully!

3

+ 5

8

(9)

Rewrite this problem as you did in #8:

97 + 28 = 125

(When rewriting make sure the 7, 8, and 5 are aligned in a vertical column; then the 2, 2, and 9 align in the tens column with the 1 in the hundreds column.)

(10)

Solve this problem. Be sure that your digits are aligned correctly: 38 x 4 (152)

(11)

Solve this problem. Be sure that your digits are aligned correctly: 6 x 42 (Write the 42 on top to make alignment easier) (252)

(12)

Solve this problem. Be sure that your digits are aligned correctly: 3289 — 97 (3,192)

(13)

Solve this problem. Be sure that your digits are aligned correctly: 9 + 4 + 2 + 11 + 8 + 51 (85; Ones digits should be aligned; tens digits, when present, aligned as well. Digits in the answer align also. There is an addition symbol only at the bottom of the column, just above the line that is drawn between the last digits in the problem and the answer.)

(14)

Solve this problem. Be sure that your digits are aligned correctly: Add together 164 and 209. From the sum of these numbers, subtract 32. (Add the first 2 numbers, aligning the ones, tens, and hundreds columns, and placing an addition sign beside the second number. The 32 may be written below the sum and subtracted, again aligning digits and using the minus sign to show the operation being performed, or a separate problem may be written to the right of the addition problem. (341)

(15)

Solve this problem, following the guidelines you have learned from similar problems:

Add together any 3 numbers of your choice. At least one of those numbers must have a digit in the hundreds place. Multiply your sum by 4. Then subtract any number you wish from your product. The only requirement is that the number you subtract must be smaller than your product. Can you explain to someone else how you chose your numbers and how you solved the problem?

(16)

Draw a picture to illustrate how you will solve this problem and then solve it. Be sure you label your answer.

A duck takes her eight ducklings for a swim. One duck dives underwater and three disappear behind a balloon that is floating in the water. How many ducklings does the mother see when she tries to count her brood? (Could be one picture, with 2 circumstances or two separate illustrations.)

(17)

Draw a picture to illustrate how you will solve this problem and then solve it. Be sure you label your answer.

Your family goes on a long trip. The first day your parents manage to drive 453 miles. The next day, because there was a water park along the way where you stopped to play, your family only travels 200 miles. On the third day your mom says that the entire trip will be 900 miles. Can you calculate how many miles you still need to travel to reach your destination? From what you have learned about your family’s travel patterns, is it likely that you can reach your destination in one more day of travel? (247; yes)

(18)

Draw a picture to illustrate how you will solve this problem and then solve it. Be sure you label your answer.

Your little sister likes to cut out paper dolls and play with them. She has 3 paper dolls and each of them has 9 outfits that fit just that doll. In addition your sister has a collection of items that will fit any of her paper dolls. She has 5 hats, 3 pairs of shoes, 6 belts and 1 bracelet. How many clothing items does your sister have altogether? (45, if you count the shoes as 6 separate items but sometimes paper doll accessories as small as shoes are cut out as a joined pair. Use the child’s illustration to guide you in evaluating the work.)

(19)

A garden is divided into thirds. In the first section there are 4 tomato plants. In the second section, twice as many pepper plants as tomato plants are growing. In the final section there are no plants growing yet. How many plants are in the garden? (12 plants)

(20)

The garden in problem 19 continues to grow, and now there are 23 carrot plants growing. However, one of the tomato plants has died and a rabbit ate one of the pepper plants. How many plants are in the garden now? (33 plants)

(21)

The garden described above is 18 feet long and 3 feet wide. How big is each of the 3 sections in the garden? The sections are all the same size. (Each section is 6 feet x 3 feet; the width remains constant for each plot and if the student drew a picture this should be obvious to him/her.)

(22)

The gardener wants to put a fence all around the entire garden. How many feet of wire will be needed? (42 feet; This is a perimeter problem; look for an indication that the child used all 4 lengths that are required to enclose the area.

(23)

A similar garden is planted against an 18-foot length of the house. Therefore, only 3 sides must be fenced. How much fencing wire is needed now? (The illustration should show 18 feet of the garden not in need of fencing; not one of the 3-foot sides.)

(24)

Sue has $3.00 to spend at the movies tucked into her pocket when she sets out to walk to the theater. She walks three blocks north from her house, then 2 blocks west. She turns left and walks 5 more blocks to reach the theater. Draw a map of the route that she took from her house to the movie theater. When she arrives at the theater, how much money does she have to spend? (Check map for directions; money does not change since the other numbers have nothing to do with spending any.)

(25)

Pretend that there is a can of pennies buried under a tree 12 blocks from your house. The tree is in a park. Write a problem that requires the finder of your map to figure out how many blocks (12) the park is from your house just from looking at your map. Then show one way to solve the problem you wrote. (Student should draw any map that indicates 12 blocks; cross streets, stop signs, or a change in house numbers are possible ways to indicate the blocks. Then the student should indicate how those blocks might be counted, or added, to reach 12.)