Roots of Reason
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Reading the Grid · Grade 4 · 4.G.A.2
Count the Steps
Name
Date
Correct / 20
1
Count the steps, then decide.
None of these sits straight on the page.
A step count says how far it is from one lettered corner to the next: how far across, then how far up or down. Count the gaps between the dots, not the dots themselves. Write both step counts, then agree or disagree with the statement under them.
1. (a)
AB goes AD goes
AB and AD meet at a right angle.
truefalse
2. (b)
EF goes FG goes
EF and FG meet at a right angle.
truefalse
3. (c)
KL goes NM goes
KL and NM run the same way.
truefalse
4. (d)
PQ goes SR goes
PQ and SR run the same way.
truefalse
Roots of Reason
Count the Steps · 4.G.A.2
Name
Date
2
Say why.
The numbers by themselves do not decide it.
5. Both pairs in 1 and 2 use the numbers 3 and 1. One pair meets at a right angle and one does not. Say what is different.
6. In 4, PQ takes twice the steps SR takes, so they are nothing like the same size. Say why they still run the same way.
Roots of Reason
Count the Steps · 4.G.A.2
Name
Date
3
Name it, then fold it.
Fold it so one half lands exactly on the other.
quadrilateral four straight sides · parallelogram two pairs parallel · rectangle four right angles · rhombus four sides equal · square both of the last two at once
7. (a)
closest name
fold lines drawn, and how many
8. (b)
closest name
fold lines drawn, and how many
9. (c)
closest name
fold lines drawn, and how many
10. (d)
closest name
fold lines drawn, and how many
Roots of Reason
Count the Steps · 4.G.A.2
Name
Date
4
One figure, every question.
Draw on WXYZ for problems 15 and 16. None of its sides runs along the grid.
11. Write the step count for each of these two sides.
WX goes
ZY goes
12. Write the angle at X using three letters.
angle
13. By its size, the angle at X is
acuterightobtuse
14. Somewhere on the outline WXYZ there is a right angle.
truefalse
15. Draw WY and XZ. Where they cross, the angle is a right angle.
truefalse
16. Draw every fold line WXYZ has.
how many
17. Write the closest name for WXYZ, then one wider name that is also true of it.
closest name
wider name
18.Triangle WXZ on its own:by its sides and by its angles
19.Triangle WXY on its own:by its sides and by its angles
Roots of Reason
Count the Steps · 4.G.A.2
Name
Date
4
One figure, every question.
Draw on WXYZ for problems 15 and 16. None of its sides runs along the grid.
20. Not one line on this figure runs along the grid lines, and you had no folded corner to test with. Say how the two step counts in 15 told you the diagonals meet at right angles.
Roots of Reason
rootsofreason.org
Reading the Grid · Grade 4 · 4.G.A.2
Count the Steps
Count the steps, then decide
1. AB is 3 right and 1 down, AD is 1 left and 3 down, true
2. EF is 3 right and 1 down, FG is 1 right and 3 down, false
3. KL is 4 right and 2 down, NM is 4 right and 2 down, true 4. PQ is 4 right and 6 down, SR is 2 right and 3 down, true
3. KL is 4 right and 2 down, NM is 4 right and 2 down, true 4. PQ is 4 right and 6 down, SR is 2 right and 3 down, true
Say why
5. In 1 the two numbers swap and one of the directions turns the other way, right becoming left. In 2 they swap and both still run to the right, so the second side leans the way the first one does instead of crossing it.
6. Which way a side runs is how far down it goes for every step across, not how far it goes in total. PQ drops 6 in 4 across and SR drops 3 in 2 across, which is 3 down for every 2 across in both.
6. Which way a side runs is how far down it goes for every step across, not how far it goes in total. PQ drops 6 in 4 across and SR drops 3 in 2 across, which is 3 down for every 2 across in both.
Name it, then fold it
7. (a) square, 4 fold lines: two corner to corner and two through the middles of opposite sides
8. (b) rhombus, 2, both corner to corner
9. (c) rectangle, 2, both through the middles of opposite sides 10. (d) quadrilateral, 0
9. (c) rectangle, 2, both through the middles of opposite sides 10. (d) quadrilateral, 0
One figure, every question
11. WX is 5 right and 2 down, ZY is 5 right and 2 down
12. angle WXY or angle YXW 13. obtuse 14. false
15. true. WY is 7 right and 7 down, XZ is 3 left and 3 down. 16. 2, the two diagonals 17. rhombus, then parallelogram or quadrilateral
18. WXZ is isosceles and acute 19. WXY is isosceles and obtuse
20. Both step counts are built from one number used twice, and one of the two directions turns the other way. That is the same pattern as 1, and it is what a right angle looks like in step counts.
15. true. WY is 7 right and 7 down, XZ is 3 left and 3 down. 16. 2, the two diagonals 17. rhombus, then parallelogram or quadrilateral
18. WXZ is isosceles and acute 19. WXY is isosceles and obtuse
20. Both step counts are built from one number used twice, and one of the two directions turns the other way. That is the same pattern as 1, and it is what a right angle looks like in step counts.
Question 7 catches a student who has only tried the folds that run at right angles to a figure's edges: they report 2 where the answer is 4. Figure (d) reaches no name past quadrilateral, and it has no fold line. Trapezoid is not on the name list, because the two definitions in wide use disagree about which figures it covers.