1. Four of the Best Numeracy focus: Problem solving focus:
Use an incomplete number square to explore patterns in the addition of four numbers
Numeracy focus:
Choose an appropriate method to add decimals; either mentally or using column addition.
Problem solving focus:
Predict, hypothesise and engage in reasoning.
You will need:
Mini whiteboards/exercise books
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2. Four of the Best Copy this number square 2 1.5 1.2 0.7 1.3 2 2.4 3.1
Choose an empty cell in the square.
Add the number (in the first row) above the cell to the number (in the first column) left of the cell.
Write your answer in the empty cell. 2
1.5
1.2
0.7
1.3 2
2.4
3.1
1.5
1.2
1.6
2.3
0.7
1.3
1.7
0.8
2.1
2.8
What strategies will you use for adding?
Click on the hyperlinked hand/calculator as an option to explore a range of appropriate numeracy strategies.
Use animation to reveal completed grid.
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3. Four of the Best
Circle one of your addition answers. 2
1.5
1.2
0.7
1.3
Cross out all the addition answers in the same row and column. 2
2.4
3.1
1.5
1.2
1.6
2.3
0.7
1.3
1.7
0.8
2.1
2.8
Choose another addition answer – one not circled or crossed out.
Repeat the above.
Keep doing this until all addition answers are crossed out.
Finally, add the numbers in each diagonal...
Add the four circled numbers...
The final animation requires students to put their mathematical ideas and observations into words. Capturing ideas and observations is an important step in investigative learning. If we move on too quickly, we may forget those early ideas which prove useful later on…
Investigation Skills
Make an observation about the totals you found.
Now add the eight numbers round the outside of the square…
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4. Ideas in the mix Four of the Best Aha! I noticed that…
The totals always
appear to…
Investigation Skills
Make an observation about the totals you found.
These sentence starters may be useful in scaffolding suitable observations.
I wonder what would happen if we changed the numbers round the edge?
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5. Try this again with this square…
Four of the Best
Try this again with this square…
Investigation Skills
Make a prediction about what may happen this time.
Say why you think this…
What do you notice about the numbers here compared
to those on the first square?
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6. Make a prediction about what may happen this time.
Four of the Best
Start again with the original square BUT this time add 0.1 to each number…
Investigation Skills
Make a prediction about what may happen this time.
Reassure students that it’s OK to be wrong when making a prediction! They’re not playing a game of ‘guess what’s in the teacher’s head’!
00.8
1.4
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7. Think carefully about your prediction this time…
Four of the Best
Does your hypothesis still work if you add 0.01 to the decimal part of each number?
Investigation Skills
Think carefully about your prediction this time…
Reassure students that it’s OK to be wrong when making a prediction! They’re not playing a game of ‘guess what’s in the teacher’s head’!
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8. Four of the Best Investigation Skills
Make a hypothesis about what may happen whenever totals are found like this in any 4 by 4 number square.
Investigation Skills
Can you write a rule or generalisation to describe what is happening?
Investigation Skills
Now test your hypothesis.
Can you find an example that doesn’t work?
Encourage students to articulate hypotheses carefully, using precise mathematical language. When trying to find an example that doesn’t work, students may not have thought about using repeated numbers, zeros or negative numbers…
A generalisation is much easier to formulate if the student has plenty of specific examples to draw on. Suggest that they complete more number squares if they struggle to say what is happening in general terms.
Clicking the ‘f(x)’ icon will take you to an optional page where algebraic notation is explored more explicitly.
Have a go at using algebra with letters in the top row and first column…
Click here to see some algebra
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9. That’s the end of this investigation.
Four of the Best
Good job!
That’s the end of this investigation.
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10. Four of the Best Numeracy focus: 1.3 + 1.8 7.3 + 1.9 1 + 1 2 0.3 + 0.8
Back to investigation
Numeracy focus:
Choose an appropriate method to add decimals; either mentally or using column addition.
‘Add the nearest multiple of 1 and adjust’
7.3
9.2
9.3 +2
-0.1
>
<
See how this number is really close to 2?
Add the 1s
1 + 1 2
Add the tenths
1.1
‘Partitioning and Recombining’
Partition
Recombine
= 3.1
If students cannot simply use place value to add numbers (e.g , ), or use number facts (e.g , ) then a reminder of the two strategies above might be useful.
The ‘Back’ icon returns you to Slide 2, where you came from.
Column addition
1.8
0.53
2.42
+1.7
_____
Line up columns carefully according to the place value of the numbers.
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11. And what is shown when terms are circled and crossed out?
Four of the Best
Back to investigation
Algebraic notation:
Don’t be scared of ALGEBRA. If anything, it can help to represent complex maths ideas a lot more simply…
What if we represent each of our 8 starting numbers with a different letter?
Now, what are our totals? a b c d e
a+e
b+e
c+e
d+e
And what is shown when terms are circled and crossed out? f
a+f
b+f
c+f
d+f
Use the animations to follow through this explanation, step by step. Students may then wish to try this for themselves, crossing out different totals to begin with.
Does this generalisation always work? What if some of the 8 starting numbers are the same, or negative?
The ‘Back’ icon returns you to Slide 8, where you came from. g
a+g
b+g
c+g
d+g h
a+h
b+h
c+h
d+h
Aha! Each of the amounts from a to h only appears once in the four circled totals…
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