B. Number · Number Sense: Big Numbers, Square Roots and Rational NumbersFinding Fractions and Decimals Between Two Numbers
B1.5 A1About 40 minutes
Learning goals- I can find a decimal between any two decimals by using more decimal places.
- I can find a fraction between two fractions by using equivalent fractions.
- I can find a number between a fraction and a decimal, or between two negative numbers.
- I can explain why there is always another number between any two numbers.
Words to know
between
Greater than the smaller number and less than the greater number.
equivalent fractions
Fractions that name the same amount, like 1/3 = 4/12.
halfway point
The number exactly in the middle of two numbers. Add the two numbers and divide by 2.
infinitely many
More than you could ever count. The list never ends.
Remember
You know how to write equivalent fractions, compare fractions and decimals, and place them on number lines. Today you will squeeze numbers in between other numbers.
The never-ending zoom
Is there a number between 0.3 and 0.4? Yes: 0.35.
Is there a number between 0.35 and 0.36? Yes: 0.355.
Between 0.355 and 0.356? Yes: 0.3555.
You can keep zooming in forever! Between any two different numbers, there are infinitely many fractions and decimals.
Strategies
For decimals: add a decimal place. Write 0.3 and 0.4 as 0.30 and 0.40. Now you can see 0.31, 0.32, …, 0.39 in between. Need more? Write them as 0.300 and 0.400.
For fractions: make equivalent fractions with a bigger common denominator. 1/3 and 1/2 have no whole-number numerator between them with denominator 6 (2/6 and 3/6). Double again: 4/12 and 6/12. Now 5/12 fits in between!
For a fraction and a decimal: change to the same form. Usually decimals are easiest.
Halfway: add the two numbers and divide by 2. This always gives a number in between.
Worked exampleExample 1: Decimals
Find three decimals between 2.7 and 2.8.
- Add a decimal place: 2.70 and 2.80.
- Any hundredths in between work: 2.71, 2.75, 2.79.
2.71, 2.75, 2.79 (many other answers work).
Worked exampleExample 2: Fractions
Find a fraction between 3/5 and 4/5.
- The numerators 3 and 4 are next to each other, so there is no fifth in between.
- Use tenths: 3/5 = 6/10 and 4/5 = 8/10.
- 7/10 is between them.
Worked exampleExample 3: A fraction and a decimal
Find a number between 0.25 and 1/3.
- Change 1/3 to a decimal: 0.333…
- So we need a number between 0.25 and 0.333….
- 0.3 works. As a fraction, 0.3 = 3/10.
0.3 = 3/10 (0.26, 0.28 and 0.32 also work).
Worked exampleExample 4: Negative numbers
Find a number between −1.2 and −1.1.
- Write them as −1.20 and −1.10.
- Halfway: −1.15. Check: −1.20 < −1.15 < −1.10 ✓
Check your answer
After you find a number, check that it really is between: is it greater than the smaller number and less than the greater number? Changing everything to decimals is a quick way to check.
Hands-on activityPaper strip squeeze
You need: a long strip of paper (cut from the long side of a sheet), a ruler, a pencil and a partner.
- Write 0 at the left end of the strip and 1 at the right end.
- Fold the strip in half and unfold. Mark the fold 1/2 = 0.5.
- Fold it in half again (into quarters). Mark 1/4 = 0.25 and 3/4 = 0.75.
- Fold once more (into eighths). Mark 1/8, 3/8, 5/8, 7/8. Each fold put a new number between two old ones.
- Squeeze game: your partner names two numbers from the strip, like 3/8 and 1/2. You name a number between them (like 7/16 or 0.45). Then your partner names a number between yours and one of the ends. Keep going for 6 rounds. Who runs out of ideas first?
What to look for: you never really run out, because you can always use more decimal places or a bigger denominator. Try to explain that to your partner.
Just for funMind-bender: there are infinitely many numbers between 0 and 1, and infinitely many between 0 and 0.001. Infinity doesn't care how tiny the space is!
Your turn
1Find a decimal between 0.6 and 0.7.
Any decimal from 0.61 to 0.69, like 0.65 (and many more, like 0.601).
2Find a decimal between 1.25 and 1.26.
Any number from 1.251 to 1.259, like 1.255.
3Find a fraction between 1/4 and 3/4.
1/2 (= 2/4). Other answers work, like 5/8.
4Find a fraction between 2/5 and 3/5.
1/2. 2/5 = 4/10 and 3/5 = 6/10, so 5/10 = 1/2 is between.
5Find a number between −0.3 and −0.2.
−0.25 (any number from −0.29 to −0.21 works, and more).
6What number is halfway between 4.6 and 4.8?
4.7. (4.6 + 4.8) ÷ 2 = 9.4 ÷ 2 = 4.7.
7Find a fraction between 1/2 and 2/3.
7/12. 1/2 = 6/12 and 2/3 = 8/12.
More practice: warm-up
1Find a number between 3 and 4.
3.5 (or 3 1/4, 3.1, ...).
2Find a decimal between 0.1 and 0.2.
0.15 (or any from 0.11 to 0.19, and more).
3Find a fraction between 1/2 and 1.
4Find a number between −1 and 0.
−0.5 (or −1/2, −0.1, ...).
5Find a decimal between 5.5 and 5.6.
5.55 (or any from 5.51 to 5.59, and more).
More practice: on your own
1Find three decimals between 0.09 and 0.1.
For example, 0.091, 0.095, 0.099. Write them as 0.090 and 0.100 to see the thousandths in between.
2Find a fraction between 3/8 and 1/2.
7/16. 3/8 = 6/16 and 1/2 = 8/16.
3Find a fraction between 1/3 and 2/5.
11/30. Common denominator 15: 5/15 and 6/15. Double to 30: 10/30 and 12/30. 11/30 is between.
4Find a number between 0.6 and 2/3.
0.65 (= 13/20). 2/3 = 0.666…, and 0.6 < 0.65 < 0.666….
5Find a number between −2 1/2 and −2.4.
−2.45. −2 1/2 = −2.50, and −2.50 < −2.45 < −2.40.
Challenge
1Is there any number between 0.99 and 1? Explain.
Yes, infinitely many. For example, 0.995, 0.999 and 0.9999. Write 0.99 as 0.990 and 1 as 1.000 to see them.
2Find three fractions between 1/5 and 1/4.
For example, 9/40, 13/60, 7/30. 1/5 = 8/40 and 1/4 = 10/40, so 9/40 is between. 1/5 = 12/60 and 1/4 = 15/60, so 13/60 and 14/60 (= 7/30) are between. Check: 1/5 = 0.2, 9/40 = 0.225, 13/60 ≈ 0.217, 7/30 ≈ 0.233, 1/4 = 0.25.
3A maple syrup recipe needs more than 3/4 cup but less than 7/8 cup. Your measuring cups go down to 1/16 cup. What amount could you use?
13/16 cup. 3/4 = 12/16 and 7/8 = 14/16, and 13/16 is between them.
Quiz
1. Which number is between 0.4 and 0.5?
2. Which fraction is between 1/3 and 2/3?
3. Which number is between 2.15 and 2.16?
4. Which number is between −0.8 and −0.7?
5. What is a good way to find a fraction between 1/4 and 1/2?
6. How many numbers are there between 1.1 and 1.2?
Big idea: Grade 7 expectation B1.5 asks students to generate fractions and decimal numbers between any two quantities. The big idea behind it is that between any two numbers there is always another number (mathematicians call this density).
Common difficulties:
- Saying there is nothing between 0.3 and 0.4, or between 3/5 and 4/5. Show how adding a decimal place or using a bigger denominator opens up room.
- Choosing a fraction with a bigger denominator, thinking it's in between (1/5 between 1/4 and 1/2). A bigger denominator means smaller pieces.
- Forgetting to check that the answer is really between.
How to help: Make the paper strip and play the squeeze game. Cooking and measuring are great real-life uses. This lesson builds A1 skills: persevering, thinking creatively and explaining reasoning.