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Number Sense Set · Sheet 1 of 7
The Bar Is a Division Sign
Name
Date
4 granola bars, 5 people. Everyone gets the same amount. Three ways to write the exact same situation:
4 ÷ 5
54
45
Same numbers, same answer. The fraction bar IS a division sign.
1. Here are the 4 bars. Cut each one into 5 equal pieces (draw the cut lines), then shade what one person gets.
One person gets of a bar.
2. Finish the chain. Each blank is the same number wearing different clothes.
45
=4 ÷ 5=
Hint: 4 ÷ 5 means 40 tenths ÷ 5. How many tenths does each person get?
Roots of Reason
Number Sense Set · Sheet 2 of 7
How Many Fit
Division can ask: how many of THIS fit inside THAT? That question works for fractions too.
1. 12 ÷ 3 asks: how many 3s fit in 12? Circle groups of 3, then count the circles.
12 ÷ 3 = (the number of circles)
2. Same question, fraction edition. 3 ÷ ½ asks: how many halves fit in 3 wholes? Each strip below is one whole, already split into halves. Count the halves.
3 ÷ ½ =
Dividing by a half gave a BIGGER answer. Hold that thought for Sheet 4.
✂ Cut-apart check. Cut out these two strips. Physically lay half-strips along the whole strip: how many fit?
1 whole
½
½
Roots of Reason
Number Sense Set · Sheet 3 of 7
×10 and ÷10, Both Directions
The digit never changes. It just slides. Left means times 10, right means divided by 10.
⟵ ×10 each step÷10 each step ⟶
hundreds
tens
ones
·
tenths
hundredths
5
.
Start: the 5 lives in the ones place. Every problem below is this same chart with the 5 in a new home.
1. Slide the 5 and write the number it becomes.
5 × 10 =
5 ÷ 10 =
5 × 10 × 10 =
5 ÷ 10 ÷ 10 =
0.5 × 10 =
50 ÷ 10 =
2. Finish the sentence with a number:
Moving one place to the left multiplies by .
Moving one place to the right divides by .
✂ Cut-and-slide. Cut out the digit tile. Slide it one column at a time across the chart above and say the number out loud at each stop.
5
Roots of Reason
Number Sense Set · Sheet 4 of 7
When Multiplying Makes It Smaller
Predict FIRST. Circle before you compute. No fair calculating and then circling.
12 × 2
=
The answer will be:
bigger than 12smaller than 12same as 12
12 × 1
=
The answer will be:
bigger than 12smaller than 12same as 12
12 × ½
=
The answer will be:
bigger than 12smaller than 12same as 12
12 × 0.5
=
The answer will be:
bigger than 12smaller than 12same as 12
12 × ¼
=
The answer will be:
bigger than 12smaller than 12same as 12
Look back: two of these rows are the exact same problem in different clothes. Which two?
12 × and 12 ×
Multiplying by a number smaller than 1 takes only part of 12. That is why the answer shrinks.
Roots of Reason
Number Sense Set · Sheet 5 of 7
One Rectangle, Four Problems
The same rectangle answers all four. Only the units change.
3 rows
4 columns
1. Each square is worth 1. The rectangle shows 3 × 4 =
2. Now each square is worth 100 (the sides are 30 and 40). Same rectangle: 30 × 40 =
3. Shade ¾ of the rectangle above. How many squares did you shade? ¾ of 12 =
4. Now the whole rectangle is worth 1, so each side is a decimal: 0.3 and 0.4. Each little square is worth 0.01 (one hundredth). The rectangle covers 12 squares, so:
0.3 × 0.4 = 12 hundredths =
That is the whole secret of decimal multiplication: tenths times tenths makes hundredths. Two decimal places, and now you know why.
Roots of Reason
Number Sense Set · Sheet 6 of 7
Fraction, Decimal, Same Number
There is no conversion rule to memorize. The bar is a division sign, so divide.
Example
12
=
21.00.5
=
0.5
2 does not fit into 1, so trade 1 for 10 tenths. 10 tenths ÷ 2 = 5 tenths.
1. Divide it out: ¾ = 3 ÷ 4
43.00
¾ =
2. Divide it out: ⅛ = 1 ÷ 8
81.000
⅛ =
3. Prediction only, no dividing: will ⅜ be bigger or smaller than 0.5? Circle one, then say why out loud.
bigger than 0.5smaller than 0.5
Roots of Reason
Number Sense Set · Sheet 7 of 7
Note Values Are the Fraction System
Every note is half the one before it. Music has been running on ÷2 the whole time.
1. Fill in the missing fractions. Each step is ÷ 2.
whole note
1
÷2 →
half note
½
÷2 →
quarter note
÷2 →
eighth note
÷2 →
sixteenth note
2. A 4/4 time signature is a fraction sitting in plain sight: 4 beats per measure, and the quarter note gets the beat. One measure below is split into quarter-note beats. How many eighth notes fit in the measure?
Eighth notes in one 4/4 measure:
3. This is Sheet 2's question with sound: how many eighth notes fit in a half note? A half note lasts 2 beats. An eighth note lasts ½ beat. Play it on the keyboard and count, then write the division it matches.
2 ÷ ½ =
(you can hear this answer)
Roots of Reason
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Number Sense Set · 7 Sheets
Answer Key
Sheet 1 · The Bar Is a Division Sign
1. Each bar cut in 5; one person gets 4/5 of a bar (one piece from each bar). 2. 4/5 = 4 ÷ 5 = 0.8
Sheet 2 · How Many Fit
1. 4 circles 2. 3 ÷ ½ = 6 halves Cut-apart: 2 halves fit in 1 whole
Sheet 3 · ×10 and ÷10
1. 50, 0.5, 500, 0.05, 5, 5 2. left multiplies by 10; right divides by 10
Sheet 4 · When Multiplying Makes It Smaller
12 × 2 = 24 (bigger) · 12 × 1 = 12 (same) · 12 × ½ = 6 (smaller) · 12 × 0.5 = 6 (smaller) · 12 × ¼ = 3 (smaller). Same problem twice: ½ and 0.5
Sheet 5 · One Rectangle, Four Problems
1. 12 2. 1200 3. 9 squares 4. 0.12
Sheet 6 · Fraction, Decimal, Same Number
1. ¾ = 0.75 2. ⅛ = 0.125 3. smaller than 0.5 (⅜ is less than ½ = 4/8)
Sheet 7 · Note Values
1. quarter ¼, eighth ⅛, sixteenth 1/16 2. 8 eighth notes 3. 2 ÷ ½ = 4
Watch Sheet 4 closely. If the prediction is wrong but the computation is right, the rule "multiplication makes bigger" is still running underneath. Reteach from Sheet 2's strips, not from more practice problems.