Justifying mathematical reasoning
Construct and follow multi-step mathematical arguments; identify errors in reasoning and explain why a method works or does not work
What a learner can do afterwards
- Explain step by step why a columnar addition method gives the correct answer
- Find and explain an error in a worked example (e.g. incorrect regrouping)
- Construct a simple argument for why a general statement is true (e.g. 'adding two even numbers always gives an even number')
1 · Read
You have explained answers with pictures, number sentences and words. Now check another person's reasoning. Compare their starting numbers with the question. Then check each calculation and why it follows from the step before. Two methods can agree because both copied the same number incorrectly. Matching answers help, but the starting numbers and steps still need checking.
In column addition, line up ones with ones, tens with tens and hundreds with hundreds. Start with ones so any carried ten is ready when you add the tens. Ten ones have the same value as one ten. For 12 ones, write 2 in the ones place and carry 1 ten. This change of grouping is called regrouping.
For 27 + 15, add the ones: 7 + 5 = 12. Write 2 ones and carry 1 ten. Add the tens, including the carry: 2 + 1 + 1 = 4 tens. Four tens and two ones make 42. Regrouping keeps the amount unchanged.
For 128 + 176, the ones give 8 + 6 = 14. Write 4 and carry 1 ten. The tens give 2 + 7 + 1 = 10 tens. Ten tens make one hundred. Write 0 tens and carry 1 hundred. The hundreds give 1 + 1 + 1 = 3. The total is 304.
A child adds 46 + 38 by writing 14 from 6 + 8, then putting 7 from 4 + 3 in front, making 714. The missing step is regrouping 14 ones as 1 ten and 4 ones. Include that ten: 4 + 3 + 1 = 8 tens. The total is 84.
Count each carry once. For 25 + 18, carry 1 ten from 13 ones. The tens are 2 + 1 + 1 = 4, giving 43. Counting that carry twice gives 53. Check alignment too. In 25 + 3, the 3 belongs under the 5 because it means 3 ones. Putting it under the 2 changes it to 3 tens.
An even number forms pairs with none left over. An odd number leaves one counter without a partner. Joining two even collections keeps every pair, so the total is even. With two odd collections, their two leftovers make another pair. With one odd and one even collection, one counter stays unpaired. Here, 4 + 6 = 10 forms five pairs.
A claim using 'always' must work in every case it names. Testing a few examples only checks those cases. A reason about joining complete pairs explains why any two even numbers have an even total. One example can disprove an 'always' claim. For example, 2 + 6 = 8 disproves 'two even numbers always make an odd total'.
To check addition another way, split the numbers: 32 + 25 = 30 + 20 + 2 + 5 = 57. Every part keeps its value. Changing the order of addition keeps the total unchanged. To check subtraction, add the answer to the amount taken away. For 85 - 27 = 58, the check is 58 + 27 = 85.
Check every step, not just the final answer, and you can find exactly where reasoning breaks down.
The lesson, practice questions and flashcards were written for this dataset.
The practice questions and readings were written for this dataset. The flashcards use standard mathematical facts. The linked videos, 'Methods for subtracting 3-digit numbers' and 'Intro to place value', are Khan Academy works. Their reuse terms have not been verified.
More to read
Read the carried digit's place
A carried 1 gets its value from the column it enters. In the tens column, it means one ten. In the hundreds column, it means one hundred.
Suppose the tens column totals 13 tens. Those tens are worth 130. Group ten of them as one hundred, leaving three tens. Write 3 in the tens place and carry 1 hundred. The amount is still 130.
When two answers agree
You are asked to add 41 + 23. You copy 23 as 33, then calculate the changed sum two ways. Both ways may give 74. They agree, but they answer a different question.
Compare the starting numbers with the question before checking the calculations. For the intended sum, 40 + 20 = 60 and 1 + 3 = 4. The answer is 64. A second method helps when its starting numbers and steps are checked too.
Name the amount counted twice
You can explain a repair by naming each amount that belongs in a column. Suppose a friend has carried one ten from the ones. The tens total needs the tens from each starting number and that carried ten once.
If your friend includes the carry twice, point to the repeated ten. Explain that the extra ten changes the total. Removing the repeated amount fixes that step. Then check the steps that follow.
Examples and reasons
Trying a sum tells you about that sum. Trying several sums tells you about those cases. A claim about every pair of even numbers needs a reason covering every pair.
Think of an even amount as complete pairs. Join two such amounts and each counter keeps its partner. No counter is left alone. This reason works whatever the two amounts are.
One example is enough to disprove a claim using 'always'. If the claim says a total must be odd, a case with an even total shows that the claim does not work every time.
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17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.