Fractions Foundations
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Path Steps
Understanding Fractions
Free PreviewExplore fractions as parts of a whole.
Why the path starts here
Fractions start with one picture: a whole cut into equal parts, some of them shaded. The bottom number counts the equal parts and the top number counts the shaded ones; every later step reads that picture.
Sets up: Comparing Same-Size Parts
Skills in this step
- Understand fractions as parts of a whole
- Identify numerator and denominator
- Read and write simple fractions
What fraction is shaded?
- The shape is cut into 6 equal parts, so the denominator (bottom number) is 6.
- 3 of the 6 parts are shaded, so the numerator (top number) is 3.
- 3 shaded out of 6 equal parts is 3/6.
Answer: 3/6
Comparing Same-Size Parts
Compare fractions with the same denominator, such as 3/8 and 5/8.
Why this step comes next
When two fractions cut the whole into the same number of parts, the one with more parts is bigger: 5/8 is more than 3/8. Comparing these first lets the child reason about size before the denominators start to differ.
Sets up: Equivalent Fractions
Skills in this step
- Compare fractions with same denominator
- Use >, <, and = with fractions
- Understand that larger numerator means larger fraction
2/8 ___ 8/8
- Both fractions have denominator 8, so the pieces are the same size.
- Compare the numerators: 2 is less than 8.
- So 2/8 < 8/8.
Answer: <
Equivalent Fractions
Discover and create equivalent fractions.
Why this step comes next
Equivalent fractions name the same amount with different numbers: 1/2 = 2/4 = 4/8. That is the tool for comparing and adding fractions whose denominators differ, so it comes before both.
Sets up: Comparing Fractions
Skills in this step
- Multiply/divide to find equivalents
- Recognize equivalence
Write a fraction equivalent to 6/12
- Equivalent fractions name the same amount. Multiply (or divide) the top and bottom of 6/12 by the same number.
- The greatest number that divides both 6 and 12 is 6.
- Divide the top and bottom by 6: 6 ÷ 6 = 1 and 12 ÷ 6 = 2, so 6/12 = 1/2.
- So 6/12 = 1/2.
Answer: 1/2
Comparing Fractions
Compare fractions using various strategies.
Why this step comes next
With equivalence in hand, two fractions with different denominators can be rewritten over a common denominator and compared part by part, exactly as in step 2.
Sets up: Adding Fractions (Same Denominator)
1/6 ___ 9/12
- Make the pieces the same size: write 1/6 in twelfths.
- Multiply the top and bottom by 2: 1 × 2 = 2 and 6 × 2 = 12, so 1/6 = 2/12.
- Now both are counted in twelfths: 2 is less than 9, so 2/12 < 9/12.
- So 1/6 < 9/12.
Answer: <
Adding Fractions (Same Denominator)
Add fractions with like denominators.
Why this step comes next
Adding starts where the parts are the same size: 2/5 + 1/5 is 3 fifths, so only the top numbers add. Learning this first heads off the common mistake of adding the denominators too.
Sets up: Adding Fractions (Different Denominators)
1/4 + 2/4 = ?
- The denominators are the same (4), so the pieces are the same size.
- Add the numerators and keep the denominator: 1 + 2 = 3, so the sum is 3/4.
- So 1/4 + 2/4 = 3/4.
Answer: 3/4
Adding Fractions (Different Denominators)
Add fractions with unlike denominators.
Why this step comes next
Adding unlike denominators combines steps 3 and 5: rewrite both fractions over a common denominator, then add the same-size parts. Neither move is new; doing them in order is.
Sets up: Multiplying Fractions
4/8 + 2/3 = ?
- The denominators are different, so rename both fractions with a common denominator, 24.
- Multiply the top and bottom by 3: 4 × 3 = 12 and 8 × 3 = 24, so 4/8 = 12/24.
- Multiply the top and bottom by 8: 2 × 8 = 16 and 3 × 8 = 24, so 2/3 = 16/24.
- Add the numerators and keep the denominator: 12 + 16 = 28, so the sum is 28/24.
- The greatest number that divides both 28 and 24 is 4.
- Divide the top and bottom by 4: 28 ÷ 4 = 7 and 24 ÷ 4 = 6, so 28/24 = 7/6.
- So 4/8 + 2/3 = 7/6.
Answer: 7/6
Multiplying Fractions
Multiply a fraction by a fraction.
Why this step comes next
Multiplying comes after adding because it means something different: 1/2 × 3/4 is “half of three quarters”. Seeing it as a part of a part explains why the answer is smaller than either fraction.
Sets up: Dividing Fractions
3/5 × 1/2 = ?
- Multiply the numerators: 3 × 1 = 3.
- Multiply the denominators: 5 × 2 = 10.
- So 3/5 × 1/2 = 3/10.
Answer: 3/10
Dividing Fractions
Divide fractions using visual models and algorithms.
Why this step comes next
Division is last because it is defined through multiplication: 3/4 ÷ 1/8 asks how many eighths fit in three quarters. The multiply-by-the-reciprocal rule is only safe once multiplying fractions is automatic.
3/4 ÷ 5/8 = ?
- Keep the first fraction, change ÷ to ×, and flip the second: 5/8 becomes 8/5.
- 3/4 ÷ 5/8 = 3/4 × 8/5.
- Multiply the numerators: 3 × 8 = 24. Multiply the denominators: 4 × 5 = 20.
- That gives 24/20.
- The greatest number that divides both 24 and 20 is 4.
- Divide the top and bottom by 4: 24 ÷ 4 = 6 and 20 ÷ 4 = 5, so 24/20 = 6/5.
- So 3/4 ÷ 5/8 = 6/5.
Answer: 6/5