Worksheet · Facts to Paper · 1 of 5
Roots of Reason
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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
Worksheet · Facts to Paper · 2 of 5
Roots of Reason
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Date
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 always gives 0.
Worksheet · Facts to Paper · 3 of 5
Roots of Reason
Name
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Neighbor Facts
Many wrong answers are the right answer to a nearby fact.
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.A student wrote 12 × 7 = 72. 72 is the right answer to a different problem. Write that problem, then explain a way to catch this mix-up before it happens.
Before you write an answer, check it against the facts just before and after it.
Worksheet · Facts to Paper · 4 of 5
Roots of Reason
Name
Date
Break Apart Multiplication
Too big to do in your head. Split it and write the pieces on paper.
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 =
Use facts you know for each piece, and keep track of the pieces on paper.
Worksheet · Facts to Paper · 5 of 5
Roots of Reason
Name
Date
Partial Quotients
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 =
Dividing a big number uses the facts you already know, one step at a time.