Worksheet · Facts to Paper · 1 of 4
Roots of Reason
Name
Date
Ones and Zeros
Four rules that look alike. Each one answers a different question.
1.Answer each one. Before you write, say the question in your head: groups of what?
7 ÷ 1 =
7 × 1 =
7 × 0 =
9 ÷ 1 =
0 ÷ 9 =
9 ÷ 9 =
12 ÷ 1 =
12 × 0 =
12 ÷ 12 =
2.Sort these five into the column that tells what happens: 6 ÷ 1, 6 × 0, 0 ÷ 6, 6 ÷ 6, 6 × 1
stays 6
becomes 0
becomes 1
3.Find the flaw. A student answered 8 ÷ 1 = 1, and did the same for ten problems in a row. Which rule was he thinking of, and why does it not apply here?
Dividing by 1 changes nothing. Multiplying by 0 erases everything.
Worksheet · Facts to Paper · 2 of 4
Roots of Reason
Name
Date
Neighbor Facts
Most wrong answers are right answers to the problem next door.
1.Solve each pair. They are neighbors, so slow down at the fence.
12 × 6 =
12 × 7 =
36 ÷ 6 =
36 ÷ 12 =
8 × 8 =
8 ÷ 8 =
8 × 12 =
96 ÷ 12 =
2.Two of these are wrong. Circle them on paper or say them out loud, then type the corrected facts below.
8 × 12 = 96
12 × 7 = 72
36 ÷ 12 = 3
5 × 9 = 30
48 ÷ 12 = 4
fix 1:
fix 2:
3.12 × 7 = 72 is a true answer to a different problem. Write that problem, then explain a way to catch this mix-up before it happens.
A near miss is a true fact in the wrong place. Check the neighbor before you commit.
Worksheet · Facts to Paper · 3 of 4
Roots of Reason
Name
Date
Break Apart Multiplication
Too big to hold in your head. Split it, and let the paper hold the pieces.
1.83 × 2. Split 83 into 80 and 3. The split is written for you: fill in the pieces.
83 × 2 = (80 × 2) + (3 × 2) = + =
2.64 × 9. Write the split yourself this time, then fill in the pieces.
64 × 9 = ( × 9) + ( × 9) = + =
3.47 × 6. Split, write every piece, add.
47 × 6 =
4.38 × 4. Same rule: every piece on paper.
38 × 4 =
Your facts do the lifting. The paper does the holding.
Worksheet · Facts to Paper · 4 of 4
Roots of Reason
Name
Date
Chunking Division
Take out easy chunks of the divisor until nothing is left. Keep count as you go.
1.96 ÷ 4. The first chunk is taken for you. Finish the job.
Chunk 1: 20 fours = 80. Left over: 96 − 80 =
Chunk 2: fours = . Left over: 0
Total fours: 20 + = so 96 ÷ 4 =
2.156 ÷ 12. Your twelves are ready for this. Start with 10 twelves.
Chunk 1: 10 twelves = . Left over: 156 − =
Chunk 2: twelves = . Left over: 0
156 ÷ 12 =
3.84 ÷ 6. Chunk it your own way. Show every chunk and every leftover on the paper.
84 ÷ 6 =
Big division is just facts you already own, stacked until the job is done.