Unit 1: Place value to a million
Usually taught in August
- Ten times as muchA digit slides one place left and becomes 10 times as much; one place right, one tenth.
- Big numbers three waysOne number in standard, expanded, and word form, period by period.
- The zeros you don’t hearFixes a mistake"Three thousand four" is 3,004, not 30,004. Zeros hold the empty places.
- Renaming numbers2,340 is 234 tens. Renaming now makes dividing easier later.
- Comparing big numbersCompare from the biggest place down until the digits differ, on a scaled number line.
- More digits, more valueFixes a mistake864 is less than 2,001, whatever the first digit says.
- Rounding: closer to which?Put the number between two round numbers on a line, mark the middle, pick the closer one.
- Rounding that rolls overFixes a mistake9,960 to the nearest hundred is 10,000: the 9 hundreds roll into a thousand.
Unit 2: Multiplication
Usually taught in September and October
- Hard facts from easy ones6 × 7 as 6 × 6 + 6; 9s as tens minus one group; 12s as tens plus twos.
- Division facts are missing factors56 ÷ 8 asks what times 8 makes 56.
- Multiplying by 10, 100, and 1,0003 × 40 is 3 × 4 tens. Each zero is a place, not a trick.
- Estimating productsRound to friendly numbers first, then use the estimate to check the real answer.
- Area model, three digits by one4 × 236 as a rectangle split into 4 × 200, 4 × 30, and 4 × 6.
- Partial products, two digits by two23 × 45 as four boxes, then the four products added.
- Three digits by two digits312 × 24 with an area model, checked with an estimate.
- The standard algorithm, two digits by two34 × 26, each step said in place value: "6 ones times 4 ones..." Required in Grade 4.
- The zero in the second rowFixes a mistakeMultiplying by the tens digit makes tens, so the second row starts with a 0.
- Where the carried digit goesFixes a mistakeThe regrouped tens sit above the tens and get added after multiplying, not before.
Unit 3: Division
Usually taught in October and November
- Sharing with blocks366 ÷ 3 by sharing hundreds, tens, and ones into 3 trays, on the sharing mat.
- Partial quotients465 ÷ 3 by taking away easy chunks (100 groups, 50 groups) and adding them up.
- Area model for divisionA rectangle with one side known, filled in place value parts.
- How to do long divisionDivide, multiply, subtract, bring down.
- The zero that holds a placeFixes a mistakeAn empty place still gets a 0 in the answer.
- Remainders as fractions7 ÷ 3 is 2 1/3: the leftover split among the 3. Florida's way.
- When the first digit is too smallFixes a mistake156 ÷ 4 starts with 15, and the answer's first digit goes over the 5.
- Estimating quotients150 tens ÷ 3 is 50 tens, so 1,500 ÷ 3 is about 500.
- What to do with the remainderFour stories, four answers: add one more van, drop it, keep only the leftover, or split it.
- 5 r2 vans is not an answerFixes a mistakeThe leftover 2 kids still need a van, so it's 6 vans.
- Times as many"3 times as many" with a bar model: one bar, then three of it.
Unit 4: Equations, factors, and patterns
Usually taught in November and December
- The equal sign means “the same as”4 × 6 = 30 − 6 is true: both sides are 24.
- Not every equation is trueFixes a mistake5 + 3 = 8 + 2 is false. The equal sign is not "the answer is".
- A letter for the unknownA story becomes 7 × n = 63, drawn as a tape split into 7 equal parts.
- Factor pairs with arraysEvery rectangle of 24 dots: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
- Prime or composite?Exactly two factors is prime. More is composite. Divisibility rules speed it up.
- 1 is not prime, and odd isn’t primeFixes a mistake1 has one factor, so it is neither. 9 is odd and composite.
- Patterns from a rule"Start at 3, add 4": extend it, then notice every term is odd.
Unit 5: Fractions
Usually taught in January and February
- Equivalent fractionsOne number line cut two ways: 2/5 and 4/10 land on the same point.
- Change both parts, not oneFixes a mistake3/4 is not 3/8. Multiply the top and the bottom by the same number.
- Comparing with 1/23/5 is more than half, 3/8 is less, so 3/5 is bigger.
- Comparing with a common denominator2/3 and 3/4 renamed in twelfths, then compared.
- Bigger numbers, not a bigger fractionFixes a mistake3/8 is less than 1/2. More pieces means smaller pieces.
- Fractions greater than one7/4 and 1 3/4 are the same amount, shown with fraction bars.
- Breaking a fraction into parts5/8 as 1/8 + 4/8, or 2/8 + 3/8: many right ways.
- Adding and subtracting like fractions3/8 + 4/8 as jumps on a number line.
- Why the denominator staysFixes a mistake3/8 + 4/8 is 7/8, not 7/16. Eighths plus eighths are still eighths.
- Adding mixed numbers2 5/6 + 1 4/6: add wholes, add sixths, then 9/6 becomes 1 3/6.
- Subtracting mixed numbers3 1/4 − 1 3/4: trade a whole for 4/4 first.
- Fraction stories, same wholeA pizza story drawn and written as an equation, both ways.
- A whole number times a fraction5 × 3/4 as five groups of 3/4, added with bars.
- Multiply only the topFixes a mistake3 × 2/5 is 6/5, not 6/15. The pieces stay fifths.
- A fraction of a number1/4 of 12 as 12 split into 4 equal parts.
- Fraction times storiesTripling a recipe that uses 2/3 cup of sugar.
- Simplest form8/12: 8 and 12 share the factor 4, so 8/12 = 2/3. 2 and 3 share only 1, so 2/3 is in simplest form. Fraction strips show 8/12 and 2/3 are the same length.
Unit 6: Decimals and money
Usually taught in March
- Tenths and hundredths6/10 = 60/100 on a 10 × 10 grid: each column holds ten hundredths.
- Fractions as decimals7/10 is 0.7 and 7/100 is 0.07, read aloud the same way.
- 0.6 is not 0.06Fixes a mistakeSix tenths is a column; six hundredths is six small squares.
- Comparing decimalsCompare from the left, or rename both in hundredths.
- A longer decimal isn’t biggerFixes a mistake0.4 is more than 0.38: 0.4 is the same as 0.40.
- Adding tenths and hundredths4/10 + 25/100: rename 4/10 as 40/100 first.
- One tenth more, one hundredth less3.9 and one tenth more is 4, not 3.91.
- Adding and subtracting decimalsLine up the places, then add like whole numbers, with money.
- Line up the decimal pointsFixes a mistake30.1 + 2.74: tenths go with tenths. Write 30.10 to see it.
- Making change$20 minus two prices, by counting up.
- Fixed and variable expensesRent is the same every month, a fixed expense; the grocery bill changes, a variable one. Then share a $10 allowance among spending, saving, and giving.
- ProfitA lemonade stand takes in $25 and spends $9 on supplies. Profit is what comes in minus what it costs: $25 − $9 = $16.
- Where to keep your savingsA jar at home, a savings account, or a bank: compare which keeps money safe and which pays interest, and what banks do with money.
Unit 7: Data
Usually taught in April
- Line plots with fractionsPencil lengths to the nearest eighth, plotted above a number line.
- Stem-and-leaf plotsTens as stems, ones as leaves, in order, with the key.
- Mode, median, and rangeOrder the data first, then find the most, the middle, and the spread.
- No mode, or manyFixes a mistakeA set can have no mode or two modes. Range is not the count.
- Answering questions from dataRead a plot, then add or subtract to answer how much longer.
- Is it a statistical question?How tall am I has one answer; how tall are the kids in my class has many. Ask a question with many answers, collect the data in a table, and say what it shows.
- Likely, unlikely, as a fractionA bag of 3 red and 1 blue marble: red is likely, blue unlikely, green impossible. Blue's probability is 1 out of 4, written 1/4.
- Timelines and Venn diagramsA bean plant grew 1/4 inch, then 1/2 inch, then 3/4 inch: a table, the same amounts as 0.25, 0.50, and 0.75, and a spreadsheet that holds the table in rows and columns and makes a bar graph. Each week goes on a timeline. A Venn diagram sorts kids who play soccer, kids who play piano, and kids who do both.
- Conclusions from dataA line graph of a puppy's weight climbs 3 pounds a month: predict about 21 pounds next month, and only a little past the data. To find the gain from month 2 to month 4, pick those two rows out of the table. Then show the same weights as a bar graph.
- Same data, different graphVotes for three lunches, tacos 20, pizza 24, and soup 22, drawn twice. With the scale starting at 0, the bars look close. With the scale starting at 18, pizza's bar looks three times as tall as tacos'. Reordered tallest first, no bar changes. The data did not change; the picture did.
Unit 8: Angles, perimeter, and area
Usually taught in April and May
- Five kinds of anglesAcute, right, obtuse, straight, reflex, tested against a paper corner.
- Measuring with a protractorCenter on the vertex, baseline on one ray, read the scale that starts at 0.
- Reading the right scaleFixes a mistakeA 60 degree angle read as 120: check against "is it less than a right angle?"
- Angles add upA missing angle on a straight line: 180 − 65 = n.
- Perimeter, area, and a missing sideA 6 by 9 rectangle, then one where area is known and a side is not.
- Area or perimeter?Fixes a mistakeAround the edge is perimeter. Covering the inside is area.
- Same area, different perimeters12 tiles in different rectangles: same area, different perimeters.
- Half a rectangleCut a 6 by 4 rectangle corner to corner: two equal triangles. The rectangle is 24, so each triangle's area is half, 12.
- Slides, flips, and turnsA triangle slides right, flips over a line, and turns a quarter turn. After each move it is the same size and shape.
- Quadrilaterals and their marksSix quadrilaterals. Tick marks show equal sides, arrows show parallel sides, and a small square shows a right angle. A kite has two pairs of equal sides next to each other and no parallel sides.
Unit 9: Measurement and time
Usually taught in May
- Measuring to the nearest eighth inchA ruler zoomed in: halves, fourths, eighths.
- Converting units with a tableFeet to inches in two columns: bigger unit, more of the smaller one.
- Multiply or divide?Fixes a mistake5 feet is 60 inches, not 5/12. Estimate which way first.
- Elapsed time on a number line9:40 to 11:15 in jumps to the next hour, across noon too.
- Time is not base tenFixes a mistake9:55 to 10:05 is 10 minutes, not 50. An hour has 60 minutes.
- Time zonesNoon in New York is 11:00 in Chicago, 9:00 in Los Angeles, and 8:00 in Anchorage. A 3-hour flight leaves Anchorage at 8:00 and lands in Seattle at 12:00, an hour ahead.
- Does that measurement make sense?A door 7 meters tall? No, a door is about 2 meters. A dog that weighs 30 kilograms? Yes. Recess that lasts 3 hours? No, about 20 minutes. Compare each to something you know.