4 (more) practical ways to teach your kids math reasoning
Help your kids understand math reasoning by asking them these questions. Why these questions are helpful, and how to use them at each grade as you homeschool math.
Mathematical reasoning is an essential skill that kids need to develop, so they can apply and use their math knowledge to a wide range of problems and make connections between different areas of math.
Reasoning helps kids make sense of math for themselves, and in our last blog post, we explored four simple ways to introduce reasoning into your math homeschooling.
In this post, we'll help you add to the range of questions you can use to help get your kids' reasoning by exploring four more question structures. For each question, we've unpacked why they are effective and given at least two examples from different grades of how it could be used in your math homeschooling.
You'll also find some top tips and things to watch out for, as well as which of your kids' reasoning superpowers they will help develop.
Hard and easy
Ask your child to give you an example of a ‘hard’ and ‘easy’ answer to a question, explaining why one is ‘hard’ and the other ‘easy’.
Why it’s effective
This question type encourages children to think closely about the structure of mathematics and enables them to demonstrate a conceptual understanding of concepts. Children enjoy the challenge of producing ‘hard’ examples that still meet the requirements set out in the question.
Children's choices when responding to this strategy often provide valuable information about what they find difficult, which may not always be what you expect! For example, if a child constantly gives calculations involving decimals as 'hard' questions, this potentially indicates they are insecure with decimal place value.
This question structure helps kids use their organizing, classifying, imagining, expressing, specializing, and generalizing math superpowers.
Examples
3rd Grade
First ask: Can you give me a shape that it'd be hard to find the perimeter of? What about a shape that it'd be easy to find the perimeter of?
Then say:
How do you measure the perimeter of a shape?
What would make measuring the perimeter more difficult?
Example answer:
Hard: A shape with lots of different length sides OR a shape with sides that are curved (as it’d be harder to measure the perimeter using a ruler).
Easy: A regular shape where all the sides are the same length.
5th Grade
First ask: Give me a hard and easy question that involves adding fractions.
Then say:
How do you add fractions?
Do the denominators of fractions have to be the same to add them?
Example answer:
Hard: 3/10+ 1/3 + 1/6 = 12/15 as it involves 3 fractions, each with different denominators, and the answer after you've converted all fractions to have the same denominator; 9/30 + 10/30 + 5/30 =24/30 as it can be simplified.
Easy: 1/6+ 4/6 = 5/6 as it only involves 2 fractions, which have the same denominator, and the answer can't be simplified.
Top tips
Kids should be encouraged to explain why the examples they have given are 'hard' or 'easy.' This could be by writing an explanation or verbally convincing a partner/an adult that their responses are 'hard' or 'easy.'
Unlike the other strategies in our previous blog, this strategy works best if your kids are encouraged to respond individually first. Once they have produced their own 'hard' and 'easy' responses, they could then be encouraged to discuss and compare their responses with your own thoughts or other kids' responses.
You can then use the 'What's the same, what’s different' strategy to encourage kids to compare their responses and draw out key concepts.
Watch out for…
Kids sometimes respond to the request for a 'hard' example by giving large multiples of 10 (e.g., 46000 + 20000=?). If this happens, ask them to convince you why this is a hard example.
Then discuss how this could be made 'easy', e.g., by multiplying/dividing by a multiple of 10 and using known facts (in the example above, 46 + 20 = 66, 66 x 1000 = 660000).
Arrange these
Give your child a set of mathematical items (for example, numbers, shapes, calculations, etc.) and ask them to either order the items or put them into groups.
Why it’s effective
This strategy encourages kids to think about the properties of the items they are given and to reason about them. It encourages them to make links between the items and to consider how they are related.
This question structure helps kids use their organizing, classifying, conjecturing, and convincing math superpowers.
Examples
2nd Grade
First ask: Can you arrange 30, 100, 19, 338, 339, 17 into an order?
Then say:
Which number is the biggest? Which is the smallest? How do you know?
Could you arrange the same numbers into groups?
Example answers:
Ordered from lowest value to highest value: 17, 19, 30, 100, 338, 339.
Grouped into odd and even numbers:
Odd numbers: 17, 19, 339
Even numbers: 30, 100, 338
Grouped into 2-digit and 3-digit numbers:
2-digit numbers: 17, 19, 30
3-digit numbers: 100, 338, 339
6th Grade
First ask: Can you arrange: 12:15, 1/4, 1:4, 1/4, 20%, 1/3, 1:2, 2/5, 80% into three groups?
Then say:
Could you place them into groups based on their value?
What do you know about the equivalence of fractions, percentages and ratios?
What could you call your groups?
Is 1:4 the same as 1/4 or 1/5? How many parts are there in total in 1:4?
Example answer:
Equivalent to one-quarter: 1/4, 25%, 1:3
Equivalent to one-third: 1/3, 33%, 1:2
Equivalent to eight-tenths: 2/5, 80%, 12:15
Top tips
This question structure works best with items that can be grouped or ordered in more than one way.
Once kids have ordered or grouped the items in one way, you could ask them to find another way to order or group them.
You can focus the use of this question structure by introducing additional instructions (i.e. can you order these angels based on their size) but make sure that the additional instructions do not remove all elements of reasoning.
You can encourage kids to generalize by focusing on what the items they have ordered have in common.
Watch out for…
Children may stick with just one way of ordering or grouping. If this happens, provide questions about specific items in the set or use the 'what's the same, what's different? strategy to help kids see the different connections between the items.
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Silly answers
Ask your kids to give you a ‘silly’ answer to a question and explain why it is a silly answer.
Why it’s effective
By asking your child to give you a 'silly' answer to a question, they will have to reason about the range in which the possible 'correct' answers could fall into. This will require them to consider the properties that the question entails and will encourage them to make a generalization about the 'correct' answer(s) in order to explain why their answer is silly.
This question structure helps kids use their imagining, expressing, specializing, and generalizing math superpowers.
Examples
2nd Grade
First ask: Give me a silly unit to measure the side of this book in.
Then say:
What units could we use to measure length?
What equipment might you choose to use to measure the side of the book?
Example answers: Feet: A foot is about the size of the edge of the table, and this book is much smaller than this.
Yard: A yard is about the width of my bed, and this book isn't as wide as my bed.
4th Grade
First ask: Give me a silly answer to the question, "What number is a factor of 36".
Then say:
What is a factor?
What do we know about the numbers which are factors of any given number?
Example answers:
39: as this is higher than 36 and factors are whole numbers that multiply together to make a number; therefore the factor of a number cannot be higher than the number itself.
7: as we know 7 × 5 = 35 and 7 × 6 = 42, therefore 7 cannot be a factor of 36.
0.5: as this is a decimal number, and factors always must be integers.
Top tips
Always ensure you ask your kids to justify their silly answer and explain why it can’t possibly be a ‘correct’ answer.
You could also ask your children to create a set number of 'silly' answers and order them in order of 'silliness.' You could then encourage them to identify which 'silly' answer is close to the 'real' answer or involves a common error/misconception.
You could also modify the question to deepen your child’s thinking and reasoning — for example, 'Can you give me a silly answer below 100?' or 'Can you give me a silly answer close to the real answer?'
Watch out for…
When asked for a 'silly' answer, children's natural instinct is often to go for an exceptionally large answer (e.g. 4 trillion, infinity, etc.). If this happens, you could ask your child if they can prove this is not an answer to the question or restrict the range of answers allowed.
What else do we know?
Give your child an 'If ...' statement (e.g., if half of 20 is 10) and ask them what else they know based on this statement.
Why it’s effective
This strategy helps kids to see the links that exist in all areas of mathematics. It also encourages them to reason and combines other known facts with the statement.
This question structure helps kids use their imagining, expressing, specializing, and generalizing math superpowers.
Examples
3rd Grade
First ask: If we know that 56 ÷7 = 8, what else do we know?
Then say:
What is the inverse of division?
Can we create two multiplication statements from this one division statement?
Example answers:
Answers linked to the inverse: 8 x 7 = 56; 7 x 8 = 56.
Answers linked to the inverse and multiples of 10: 70 x 8 = 560; 700 x 8 = 5,600.
Answers linked to related division facts: 56 ÷8 = 7; 560 ÷70 = 8 ; 5,600 ÷70= 80.
7th Grade
First ask: If we know the radius of a circle is four inches, what else do we know?
Then say:
How is the radius of a circle linked to its diameter?
How can we find the area of a circle?
Example answers:
The diameter of the circle is 4 in.
The circumference of the circle is 25.13 in.
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Top tips
Provide the statement and allow your kids to record everything else they know. Adding a time and/or quantity challenge (e.g., can you state at least ten other facts in two minutes?) can help add an additional challenge, element, or competition!
You can also help your kids identify and practice the 'automatic' related facts that they should be able to state almost instantaneously, e.g., inverse facts (7 x 13 = 13 x 7) and multiples of 10 (70 x 8, 7 x 80, etc.).
Watch out for…
Sometimes children will produce a few 'obvious' related facts (for example, using inverses, etc.), but then struggle to see any other related facts. If this happens, you can encourage them to combine related facts to create new facts, e.g., use an inverse while also dividing by 10 (so 0.3 x 6 = 1.8 is related to 6 x 3 = 18).
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