3rd Grade - Be Able To Generalize Results

Be Able to Generalize Results
Develop generalizations of the results obtained and apply them in other circumstances.
Mathematical Reasoning: develop generalization of results, problem solving The answer or process to solve a problem is similar to another problem or situation and can therefore be applied to the new problem or situation, as well.

Sample Problems


If you know that you can find even numbers by skipping every other number and that’s also counting by 2s, such as 2, 4, 6, 8, can you apply 1010, 1012, 1014, etc? (Yes.)


If it gets dark at 8pm, is it reasonable to say that it will get dark tomorrow at 8pm? (Yes)


1 x 11 = 11, 2 x 11 = 22, 3 x 11 = 33 Do you see a number pattern? (any number 1-9 times 11 is that number doubled)


What is the pattern? 23, 27, 31, 35 (+ 4)


What is the next number? 95, 100, 105, 110, 115 _____ (120; +5)

Learning Tips


If you’re a person who likes order and enjoys tasks that require organization, you are aware of how important a sense of order is to creating a lifestyle that minimizes chaos. Third grade children often do not yet have a sense of how much of our life rhythms revolve around patterns—from the regularity of our world’s light/dark rhythms to the predictable schedule of school days, offset by the more freestyle days of summer and vacations. It is important to point out these patterns to your child in daily conversation, planning, and activities. Not only will this sense of order help them mathematically, it will also assist them in daily living.


When you work on math problems with your child take every opportunity you can to help children see the patterns that are developing in their increasing understanding. For example, “skip-counting: of which counting by 2’s is the most common but it is possible (and desirable) to skip-count by 3’s, 4’s, 5’s, 6’s, 7’s, 8’s, 9’s, 10’s, and possibly even 11’s and 12’s shows children that there is order and a recognizable pattern to the way we group like-sized groups of items so they can be quickly counted and combined. When children see that there are really only a few types of word problems and that all problems are based on some slight variation of one of those types, their anxiety about the amount of material that must be learned diminishes. Similarly, when you can show them that because, for example, 8 x 7 is really the same problem as 7 x 8, so in reality there are many fewer multiplication facts to memorize, children can see that noticing the patterns they encounter and figuring out a way to use them will, in the long run, save them time and effort.


So that your children do not use memorizing how to do every problem they encounter as their primary strategy, guide your children only until they can see the pattern involved. Help them to become independent as soon as they can manage it.


Once your child can analyze and generalize about number patterns they can use this information to create a “rule” for themselves or make a chart that summarizes their findings. A small notebook where these truths can be written will create a valuable resource for your child to use in the future. More will be gained if the child does the writing!


Often children for whom math seems “easy” are really children who quickly see and use the mathematical relationships and patterns. If your child isn’t “naturally” one of them, the skills involved can be taught. It’s mostly a matter of experience with patterns and relationships that leads to a greater understanding of them.


Of course, some relationships that seem to be patterns will turn out not to be so. Two common generalizations that children make (and are sometimes taught by adults who haven’t thought it through either!) are: “When writing numbers all numbers open to the left “(5 has a right opening and a left opening) and many numbers have no “opening” at all, as conventionally written) and “problems with big numbers will be hard” which isn’t so either; some of the trickiest involve very small numbers! If you see your child falling into such a trap—even if it is just the observation that could indeed be true in one instance that “all of the even problems on this page require me to regroup numbers and the odd ones don’t” point out that here is a situation of coincidence, not patterning. Such a convenience may well not occur again.


The Extra Help Problems have been designed to help the learner discover more valid mathematical patterns than perhaps he/she already knew.

Extra Help Problems


Three tulips grow in the flowerpot each spring. My mother puts the bulbs in the refrigerator until the next year. Is it an accurate generalization to assume there will be three tulips again next year? (Yes.)


If 25% of the classroom likes burgers and 50% likes pizza and 25% likes burritos, how many children like those things if the whole school is surveyed? The school has 100 children. (25, 50, 25)


Find the perimeter around a combined shape. Show a picture of a square with 2 inches on each side and a triangle with 2 inches on each side. Put the shapes together. What is the perimeter around the new shape? (10 inches)


In this sequence of numbers, what is the pattern? What will the next number in the pattern be? 0, 5, 10, 15, ___ (20; the pattern is counting by 5’s)


In this sequence of numbers, what is the pattern? What will the next number in the pattern be? 100, 92, 84, 76, 68, __ (60; 8 is subtracted from the end number each time)


Mary completed a word problem, which asked her to state how much change she would receive from a $5 bill if the 3 items she bought each cost $1.10. She solved the problem by adding up the cost of her items and then subtracting that value from $5.00. What parts of her solution form a pattern that she can remember and use the next time she sees a problem like this? What parts of the problem will change and are therefore, not part of the pattern? (The method of solution’s pattern is that one can add up the value of the items and subtract the total from the available money. The parts of the problem that will not be useful next time are how many items, how much they cost, and how much money she had to spend.)


Often there is more than one pattern that can be used to complete a problem. Can you think of another one for problem 6? (Mary could subtract the price of one item from $5.00, then subtract the price of the next one from that value, etc. until all of the purchased items are accounted for.)


Which of these patterns do you think is more useful? Give reasons for your choice. (It depends on the numbers involved. If the prices are all easily subtracted—perhaps they all end in a 0 and there aren’t many of them, the second pattern might be easiest and most accurate. If there are many items, and the prices end in odd-cents, the first pattern is probably faster and more accurate.


What addition pattern do you see in these solutions?




8 +8=16



A pattern so strong that it’s a rule: “When 2 even numbers are added together, the sum is always an even number.”


What addition pattern do you see in these solutions?





A pattern so strong that it’s a rule: “When two odd numbers are added together, the sum will always be an even number.”


Can you write 6 examples that demonstrate this rule?

When an odd and even number are added together the sum is an odd number.” (Accept any set of combinations that follow the rule and are correctly solved.)


Look at a multiplication rubric. (See link in the online resources.) What patterns do you see? (Skip-counting patterns, several rules:

  1. If both multiplicands are even, the product is also an even number.

  2. If only one multiplicand is even, the product is still an even number.

  3. If both multiplicands are odd, the product is always an odd number.


Jose wrote these multiplication problems and solved them. He noticed a pattern that will save him time in the future. What is that pattern?

7 x 1 = 7


70 x 10 =700

700 x 10=7000

(When we look at the first two problems, multiplying by 10 only changes the place values. In order to move the 7 to the tens place, we only need to add a 0. Multiplying by 100 requires moving the 7 to the hundreds place, requiring 2 zeros to place the seven correctly. A short-cut “rule” may be written: “Multiply the non-zero digits as always. Count up the number of final zeros. Add that many total zeros to the product you calculated earlier. The rule works only for final digits, not internal ones; i.e., 701.


Maribel wonders how to change a whole number into a fraction so that she can use that number to solve problems. She decides to figure it out for herself. She draws several pies. She divides the first pie into 3 pieces. Using the rule she learned that says the bottom number tells how many pieces a pie is cut into, she writes down /3. Then she uses the rule that the top number says how many pieces are still in the pie, she writes 3/3 since no pieces are missing. Then she draws another pie and divides this one into seven pieces. Following the same rules, she expresses this whole pie as 7/7. Then she draws a pie and divides it into 24 tiny pieces and expresses that pie as 24/24, following the rules she knows. What rule can Maribel write now that will tell her how to express any pie as a fraction, using any denominator that she wants to use? Draw some pies to explain your answer.

(Rule: To express one (or a whole pie), use the same number for the numerator as the denominator)


A group of numbers in a relationship allow us to use that relationship to show the pattern that exists between them These numbers are in an relationship: 3, 4 and 7. Write all of the relationships they have. What pattern do you see?





Pattern: The two smaller numbers, added in any order, are of the same value as the larger number. Either one of the smaller numbers, subtracted from the larger number leaves the other small number. No other number can be substituted for one of these numbers and still have a valid relationship.


Write the number sentences that show the relationships between 9, 8, 17.

(9+8=17; 8+9=17; 17-8=9; 17-9=8)


Write the number sentences that show the relationships between 15, 9,and 6.

(9+6=15; 6+9=15; 15-6=9; 15-9=6)


Write the number sentences that show the relationships between 2, 3 and 6. (This is a multiplication/division pattern relationship:

2 x 3=6

3 x 2=6

6 / 3=2

6 / 2 =3


What pattern do you see among the numbers 72, 9 and 8?

(A multiplication/division pattern; the number sentences can only be:

If there are 8 items in each of 9 groups, there are 72 items in all.

If there are 9 items in each of 8 groups, there are 72 items in all.

72 divided so that 8 items are in each group will produce 9 equal groups

72 divided so that 9 items are in each group will produce 8 equal groups

Or, the number sentences can be written traditionally.


Multiply several different numbers by zero. Look at your answers. What pattern do you see when a number is multiplied by zero?

(No matter how big the number is, if you multiply is by 0, the answer is always zero. This is because no matter how many groups you have, if there is nothing in them, you still have nothing.)


Wilhem is trying to finish his math test before the bell rings. He is almost finished but groans when he sees the final problem. One of the numbers is so big! How can he finish working any problem when one of the numbers in the problem has 17 digits! Then he notices that he has been given a division problem. He is to divide this 17-digit number by zero. Wilhem sighs happily, writes down the answer and turns in his paper in plenty of time. Why was Wilhem able to so easily finish his test when the final problem looked so time-consuming?

(Wilhem remembered the pattern rule that says, “You can’t divide something by zero. If you have one item, or a million items, you can’t divide them into no groups. You simply can’t divide them with those conditions! Therefore Willem wrote down, “Can’t be solved” and was finished.)


Use the zero-pattern rules you have learned to solve:

3,184 x 0 = ? (0)


0 x 27,493,130 = ? (0)


10 x 2,487 = (24,870)


100 x 4,386,149,302= (438,614,930,200)


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