This is the third blog post in a series about building number sense foundation.
I want to ensure that you have read the previous two as this post assumes you understand why you need to go ALL the way back to where students currently are to build that solid foundation as well it assumes that you have worked on subitizing, benchmarking to 5 & 10 and partitioning.
As with the other number sense basics, this skill may seem really obvious to you, but as with the other basics, I see a whole lot of students who don’t automatically know the relationships between numbers in this way. For students who already have this skill, you can extend this to finding 10 or 100 more/less (looking for similarities and differences between finding one/two more and less and finding 10 or 20 more/less). For example, if a student is working on explaining that 10 less than 54 is 44 but says that they take 1 away from the 5, then they are not using proper place value language and so are likely not thinking in place value but rather in digits. To build number sense (not digit sense) we need students to be very aware of place value.
Just like with all of the other concepts visuals are key here. It is really valuable for a student to literally see on a 10 frame that 7 is two greater than 5 or that 5 is 2 less than 7 (students need more practice with LESS than because it is used less often in daily life). When they are just figuring these relationships out the visuals along with the numerals will help them to make the needed connections. Number paths or number lines can also be used but eventually, we want them to know these facts and so we need to give them plenty of time to practice.
To give students a chance to practice this skill you can give them dice (I like to use different sided dice to differentiate – see the picture below for examples of different sided dice). Then students roll their die and say what 1 more and 1 less of the number is, before passing it to their partner. The partner ensures that it is correct before they have their turn. You can do the same activity for 2 more and less once they’ve mastered 1 more and less. If you spent some time with the 10 frames when working with subitizing and again when working with partitioning then they should be pretty familiar with how numbers relate to 5 and 10 so now we are just going to be more explicit about it. You can ask questions for 8 like: “What is 1 more? What is 2 more? What is 1 less? What is 2 less? How do you know? Is it closer to 5 or 10, how do you know?” For those students who are still counting up or back by 1 or 2 when asked to do this, they should have a 10 frame so that they can start seeing the numbers and use more of their subitizing strategies rather than relying on counting. Number lines are a great visual too.
For those students that need more of a challenge, you can give the larger sided dice and ask them to discuss the place value connections they are making. For example: if a student knows without counting that 8 is 2 less than 10 but then when they roll 30 and they don’t know that 28 is 2 less without counting you can use 10 frames to show them the connection between 8 and 10 as well as 28 and 30 (you can print filled in and partially filled in 10 frames that are great to use with larger numbers here from Carole Fullerton – she also has Canadian money that can be used as an another visual to work on the same skill:
This concept is the same when we add or subtract 10 or 100 to a number. We want to be explicit about the place value rather than using tricks like add one to the tens place. We want to actually ensure they understand that they are adding 10 or 100, not 1.
Rounding is one of those concepts that we often teach by telling students “look to the immediate right of the place value you are rounding and if it’s 5 or greater roundup and if it’s less than 5 round down”. Unfortunately, this doesn’t help them to understand what it means to round AND this rule only works in base 10. Considering that in our Canadian money, we now round to the nearest nickel (base 5), this method does not work at all.
Instead of this rule, use manipulatives or number lines to show that what we are really doing when rounding is figuring out what number we are closest to. It is just that simple. For example:
Round 437 to the nearest 10. Let’s look at this on a number line:
It is obvious when we see it that 437 is much closer to 440. So, what happens with 435? Because it is the exact middle it is an important number to explore and discuss. We can let our students know that by convention we round UP when we find ourselves at this midpoint but, in reality, it isn’t close to either number because it is in middle. For all other numbers, it is better to think about it as ‘what is it closer to’.
This same concept applies for decimal and fractional numbers too:
We can also use base 10 blocks to help us round in terms of ‘what is it closest to?’
You can give your students some tasks such as:
Round 1 497 to:
a) the nearest thousand _____________
b) the nearest hundred ______________
c) the nearest ten ____________
d) the nearest five ___________
I hope you found these 3 blog posts useful. Although it is challenging to go so far back when we have such a range of abilities and skill levels in our classes, the students who are missing these skills need them to build a strong foundation. The students who already have these skills can be engaged in other forms of problem-solving, logic puzzles (solving puzzles and playing games are curricular competencies). There are great problems at this site: www.nrich.maths.org that will challenge your ‘at grade level’ learners while you spend some time on these basics. Remember that these activities can also be sent home as parents can help to support these basics too.
Here are some freebies for you to help you along with working on these skills. I want to write blogs and vlogs that will help you, please send me topics you would like to see posts about or questions you have or add them to the comments below.
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